How to arrange fractions in ascending order. Topic: "Comparing fractions with different denominators

A fraction is a ratio of two numbers that can be used to represent any element of a rational set. According to the method of writing, fractional numbers are divided into ordinary types m / n and decimal. Ordinary fractions with different numerators and denominators are difficult to sort in ascending / descending order on an intuitive level, as is the case with decimals. That's what our calculator is for.

Representation of rational numbers as a fraction

When people faced the problem of separating a part from a whole, they came up with fractions. If you divide the round cake into 4 pieces, then each piece of treat will be 1/4 of the whole cake. With the introduction of the decimal system, 1/4 turned into 0.25, and for modern people this designation of the fourth part of something is much clearer. However, 0.25 can be expressed in an infinite number of fractions: 1/4, 2/8, 25/100, or 752/3008. The last fraction is not at all obvious and it is not intuitively clear what number it represents.

This problem also arises in cases where there are many different fractions in front of your eyes. Finding out which fractional number is greater or less at first glance is very difficult: you have to calculate the ratio of numbers in your mind or bring them to a common denominator. Depending on the presented set of fractions, their sorting occurs in different ways.

Fractions with the same denominators

Sorting such fractions is not difficult. If rational numbers have the same denominator, then they are ordered by numerators. For example, for the set 1/5, 10/5, 4/5, and 3/5, it is obvious that the elements are sorted:

ascending - 1/5, 3/5, 4/5, 10/5;
descending - 10/5, 4/5, 3/5, 1/5.

The main rule: look at the numerators and sort by them.

Fractions with the same numerator

The set of rational numbers may look different: the denominators are all different, but the numerator is the same. For example, we have a set: 3/5, 3/20, 3/10, 3/7. How to sort them? In all cases, we divide the three into different numbers, and the larger the denominator, the smaller the value of the fraction. Obviously, the number 3 divided by 20 is in any case less than 3 divided by 5. If we calculate these values, we get decimal fractions 0.06 and 0.6, and such values are not difficult to compare. Sorting such fractions is performed by denominators, but in reverse order. For our example, sorting would look like this:

ascending - 3/20, 3/10, 3/7, 3/5;
descending - 3/5, 3/7, 3/10, 3/20.

The larger the denominator, the smaller the value of the fraction. The main rule: look at the denominators and sort the numbers in reverse order.

Completely different fractions

The previous examples were too simple. In most cases, sets of rational numbers contain completely different fractions, with different numerators and denominators. In this situation, the only correct sorting method is the method of casting all elements to a common denominator. There are three methods for determining the common denominator: using the maximum denominator, iterating over multiples, or factoring. In the general case, the search for a common denominator is reduced to the task of determining the least common multiple ().

The first method involves checking the largest denominator for divisibility by the rest. If the maximum denominator is divisible by the remainder, then it is multiplied by 2, 3, 4 and so on until it becomes a multiple of all other denominators. The second method is more difficult, since we need to sequentially write out multiples for each denominator until there are common ones, which is also inconvenient.

The most convenient, and therefore the most common method of finding the LCM is to factorize. Every integer can be factored into prime factors in a unique way, up to the order of the factors. For example, the number 30 can be decomposed into 2 × 3 × 5, and the number 20 into 2 × 2 × 5. The least common multiple of these numbers is the number that consists of the indivisible factors common to these numbers. For this pair, it is 2 × 2 × 3 × 5 = 60.

Carrying out these operations manually is a long and tedious task. Our program automatically sorts common and decimal fractions in ascending or descending order. To do this, you just need to enter the values \u200b\u200bseparated by a space into the calculator form and make one click with the mouse. The peculiarity of the program is that in the case of a heterogeneous set of rational numbers (decimal and ordinary fractions), the calculator first sorts decimals, and then ordinary fractions. Thus, the calculator divides the mixed sets into two sets of common and decimal fractions and sorts them separately.

Consider an example

Sorting example

Suppose we have a collection of heterogeneous numbers:

1/5, 2/9, 0,75, 5/7, 0,2, 6/13, 0,35, 8/15.

At first glance, you can’t guess which of these numbers is the largest and which is the smallest. Manually, we would have to factor or select multiples, but with the help of a computer, we can choose from:

convert ordinary fractions to decimals;
sort them using an online calculator.

Let's try both. Let's represent our population in the form of decimal fractions:

0,2 0,22 0,75 0,71 0,2 0,46 0,35 0,53

We simply calculated the value of the given fractions and arranged them accordingly to the original series. Sorting such numbers is as easy as shelling pears, but again, this is an extra effort for intermediate operations. Let's just enter our series into the calculator form and get the answer:

ascending - 1/5, 2/9, 6/13, 8/15, 5/7; 0.2; 0.35; 0.75;
descending - 0.75, 0.35, 0.2; 5/7, 8/15, 6/13, 2/9, 1/5.

Conclusion

Sorting fractional values is necessary when processing any data, so in practice you may encounter the need to order different values. For students, our calculator is useful for checking solutions in arithmetic.

Sections: Mathematics , Elementary School , General pedagogical technologies

Purpose: creating conditions for comparing fractions with the same numerators and different denominators through the inclusion of students in an educational study.

1. Face a problem on the topic of the lesson and find a way out of it;

2. They will derive a rule for comparing fractions with different denominators and the same numerators;

3. Learn to compare such fractions;

4. Continue the formation of communicative relations.

STUDY PROCESS

1. Org. moment.

2. Actualization of knowledge.

Sort the numbers into groups

134, 58, 632, , , 178, , 245, , 11, 6.

(The numbers are written on the cards).

How did you distribute the numbers?

(Whole numbers, fractional numbers -

134, 58, 632, 178, 245, 11, 6.

Arrange these fractions in ascending order.

And how did you know that the fractions had to be arranged like that?

( - the smallest fraction, - the largest fraction).

Conclude: If a fraction has equal denominators and different numerators, then the fraction with the larger numerator will be larger.

Post the rule on the board.

And now I suggest you compare these fractions. Consider them.

What did you notice? (The denominators of the fractions are different, the numerators are the same).

Find among these fractions the smallest and largest?

There have been many opinions. We have a problem:

How do you compare fractions with different denominators?

To answer the question, we will conduct research work.

We will work in groups according to the instructions.

Instruction

Consider the numbers carefully.
Place these fractions on the coordinate ray, on the selected unit segment.
Compare the obtained segments. Make a conclusion.
Arrange the fractions in ascending order. Highlight the small fraction in green and the large fraction in red.
Try to formulate a conclusion: how to compare fractions with different denominators.

Group Report

I group. We compared the fractions and arranged them in ascending order like this (on the fraction cards)

What is your conclusion? (The larger the denominator of the fraction, the smaller the fraction with equal numerators).

Each group reported and made their own conclusion.

On the board are strips of children of each group with fractions arranged in increasing order.

What is the smallest fraction among all fractions?

How can we choose?

Compare the reports of each group.

What did you notice?

The same fraction is marked with a different color. Why? (They compared among different fractions).

What order are we in?

(In ascending order

What is the smallest fraction? ()

And which is the biggest?

We can now answer the question of how to compare fractions with the same numerators and different denominators. What is the rule?

Make a general conclusion:

For fractions with equal numerators, the larger the denominator, the smaller the fraction.

Let's compare our findings with scientific ones.

Read p.43 from the textbook.

What have we learned to do today?

This was the topic of our lesson.

Hang out.

Now try to arrange the new fractions in ascending order. No. 101(5)

What should we pay attention to?

(numerators are the same, denominators are different)

To arrange fractions in ascending order, you need to find the fraction with the largest denominator and arrange them in descending order.

3. The result of the lesson.

What new did we learn at the lesson today?

What did you learn in class?

Homework: come up with a diagram for easy comparison of fractions.

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Goals: put a problem on the topic of the lesson and find a way out of it; derive rules for comparing fractions with different denominators; learn to compare fractions with different denominators; continue building relationships.

Information for the teacher In the course of completing assignments during all lessons, students pronounce the rules for comparing, reducing, adding and subtracting ordinary fractions, and formulate the main property of a fraction.

I. Organizational moment

II . Updating the basic knowledge of students

1. To acquaint students with the results of independent work.

2. Solve the tasks where the greatest number of errors were made.

III. Verbal counting

1. Name some numbers that have only three divisors. What pattern can be seen? (9, 25, 49, 81 are the squares of natural numbers, the numbers themselves are odd.)

2. Cut:

3. Bring the fractions to the lowest common denominator:

4. The teacher checks all notebooks in 22 minutes.

What part of the notebooks will the teacher check in 1 minute? in 9 minutes? in 16 minutes?

5. A full fruit box weighs 22 kg. A half-filled box weighs 12 kg. How much does an empty box weigh?

Solution:

1) 22 - 12 \u003d 10 (kg) - half the fruit weighs.

2) 12 - 10 = 2 (kg).

(Answer: an empty box weighs 2 kg.)

IV. Individual work

1 card

1. Reduce the fraction 2/3 to the denominator 9, and the fraction 32/40 to the denominator 5.

2 card

1. Reduce the fraction 8/9 to the denominator 18, and the fraction 56/72 to the denominator 9.

2. Bring the fractions to the lowest common denominator:

V. Presentation of the topic of the lesson

Today in the lesson we will compare fractions with different denominators.

VI. Actualization of students' knowledge

And now let's remember how fractions with the same denominators or with the same numerators are compared.

1. Divide the numbers into groups:

How did you distribute the numbers?

(Answer: for 2 groups:

integers: 58; 178; 245;

fractional numbers:

into 3 groups:

integers: 58; 178; 245;

ordinary fractions:

decimals: 13.4; 0.32; 11.6.)

Arrange these fractions in ascending order.

And how did you know that the fractions had to be arranged like that?

What is the rule for comparing fractions? (Of two fractions with the same denominator, the larger fraction is the one with the larger numerator.)

2. Write the fractions in descending order:

What does it mean to write fractions in descending order? (From largest number to smallest number.)

How do you compare fractions with the same numerator? (Of two fractions with the same numerator, the fraction with the smaller denominator is larger.)

Solution:

VII. Learning new material

1. Preparatory work.

And now I suggest you compare fractions. Consider them.

What did you notice? (The denominators and numerators of fractions are different.)

Find among these fractions the smallest and largest.

There have been many opinions. We have a problem: how to compare fractions with different denominators?

To answer the question, we will conduct research work. We will work in groups according to the instructions.

(Write instructions on the board.)

Instruction:

1. Look carefully at the numbers.

2. Place these fractions on the coordinate beam, choose a unit segment yourself.

3. Compare the resulting segments. Make a conclusion.

4. Arrange the fractions in ascending order. Highlight the smallest fraction in green and the largest in red.

5. Try to formulate a conclusion: how to compare fractions with different denominators.

Tell me, is it convenient to mark them on the coordinate beam every time you compare fractions?

How to compare such fractions?

Formulate a rule for comparing fractions with different denominators and numerators.

2. Work on a new theme.

Compare the fractions 2/3 and 3/5.

Let's reduce fractions to the smallest common denominator. (Since 3 and 5 are relatively prime numbers, the NOZ of fractions will be their product.)

3. Textbook, p. 50 (in some textbooks there is a typo - instead of the word "dative" it should be written "genitive").

Read the text under the heading "Speak correctly."

Read the record data in two ways:

(Ten fifteenths is greater than nine fifteenths, or ten fifteenths is greater than nine fifteenths.)

VIII. Physical education minute

IX. Consolidation of the studied material

1. No. 304 (a, b) p. 50 (a strong student explains at the blackboard, the rest are in notebooks).

Solution:

a) Compare the fractions 2/3 and 8/21.

Let's reduce fractions to the smallest common denominator. (Since 21 is divisible by 3, the NOZ fractions will have a larger denominator of 21.)

How do you compare fractions with the same denominator? (Of two fractions with the same denominator, the larger fraction is the one with the larger numerator.)

b) Compare the fractions 4/15 and 2/5.

Let's reduce fractions to the smallest common denominator. (Since 15 is divisible by 5, the NOZ fractions will have a larger denominator of 15.)

2. No. 305, p. 50 (write down the decision in a shorter way, pronounce the entire explanation).

Solution:

(Answer: a) 1/30; b) 9/14.)

X. Independent work

Mutual verification. Answers on the board.

Option I . No. 311 (a, b) page 51, No. 352 (a) page 56.

Option II. No. 311 (c, d) p. 51, No. 352 (b) p. 56.

XI. Working on a task

I. No. 313 p. 51 (at the blackboard and in notebooks).

Read the task.

What needs to be done to answer the question of the problem? (Compare fractions.)

Solution:

(Answer: pictures take up more space in the book.)

2. No. 315 p. 51 (at the blackboard and in notebooks).

What is known about the problem?

What do you need to know?

What shall we take as a unit? (All work.)

Solution:

Let 1 be all work.

What part of the pool is filled by a narrow pipe in 1 hour? 1/10 (part).

What part of the pool is filled by a wide pipe in 1 hour? 1/4 (part).

What part of the pool is filled by a narrow pipe in 7 hours? 7/10 (pool).

What part of the pool is filled by a wide pipe in 3 hours? 3/4 (pool).

Which pipe gives less water?

(Answer: narrow pipe.)

3. No. 355 p. 56 (after parsing it yourself).

What type of task is this task? (To combinatorial ones.)

What lesson can be the first lesson? (Any of five.)

What lesson can be the second lesson? (Any of the remaining four.)

What lesson can be the third lesson? (Any of the remaining three.)

What lesson can be the fourth lesson? (Any of the remaining two.)

What lesson can be the fifth lesson? (Only one lesson.)

What rule will we use when solving the problem? (Product rule.)

Solution:

5 4 3 2 1 = 120 (options).

(Answer: 120 options.)

XII. Repetition of the studied material

No. 281 (b) p. 46 (oral with detailed commentary).

Solution:

XIII. Summing up the lesson

How do you compare fractions with the same denominator?

How do you compare fractions with the same numerator?

How do you compare fractions with different denominators?

Homework

Topic: "Comparison of fractions with different denominators"

Item: Mathematics.

Lesson type: lesson learning new material .

Educational and methodological support:, etc. Mathematics 6th grade. Moscow, Mnemosyne, 2007

Goals: derive rules for comparing fractions with different denominators; Learn to compare fractions with different denominators.

Tasks:

Educational: learn to apply the algorithm for comparing fractions with different denominators, continue to develop the ability to reduce fractions.

Developing: develop logical thinking, the ability to draw conclusions, generalizations, develop cognitive activity, form the stability of attention.

Educational: to educate students in accuracy, a culture of behavior, a sense of responsibility, to instill interest in the subject.

Equipment: interactive whiteboard, multimedia projector, presentation, cards for self-study.

Lesson structure:

Organizational moment (2 min); Oral counting (5 min); Learning new material (15 min); Physical education (2 min); Independent work (7 min); Work on previously covered material (10 min); Summing up the lesson (2 min); Homework (2 min).

During the classes:

I.Organizational moment (2 min).

What topic did you work on in previous lessons? (Bring fractions to the lowest common denominator.)

What difficulties did you encounter? What help do you need from a teacher?

II.Oral counting (5 min).

1. Name some numbers that have only 3 divisors. What pattern can you notice? (9,25,49 ... are the squares of natural numbers, and the numbers themselves are odd)

2. Reduce fractions: ; ; ; https://pandia.ru/text/79/575/images/image005_65.gif" width="21" height="41 src="> (slide 2).

3. Bring the fractions to the lowest common denominator:

a) and https://pandia.ru/text/79/575/images/image008_47.gif" width="21 height=41" height="41">.gif" width="21 height=41" height= "41">; 0.32; 178; ; https://pandia.ru/text/79/575/images/image013_39.gif" width="16" height="41 src=">.gif" width="21" height="41 src=">. Arrange in descending order? Why? (slide 7)

-) And now I suggest you compare fractions; ; ; https://pandia.ru/text/79/575/images/image007_58.gif" width="16" height="41 src=">.(slide 8)

-) What did you notice? (denominators and numerators are different)

-) Find among these fractions the smallest and largest fraction.

-) There are many opinions. We have a problem: how to compare fractions with

different denominators?

-) To answer the question, let's do a little research work. I

I give you instructions and we will carry out tasks according to it.

Instruction: (slide 9)

1. Draw a coordinate ray, take 12 cells as a single segment.

2. Place these fractions on the coordinate line.

3. Arrange the fractions in ascending order and write them down.

4. Highlight the smallest fraction in green and the largest in red.

-) Draw a conclusion, how to compare fractions with different denominators?

-) Tell me, is it convenient each time, comparing fractions, to mark them on the coordinate beam?

-) How to compare fractions? (reduce fractions to the lowest common denominator, and then compare fractions with the same denominators using the rule)

-) Compare fractions and (slide 10).

IV.Physical education (2 min).

No. 000 (a, b) p.50, No. 000

v.Independent work (7 min).

No. 000(a, b), 352(a)

VI.Work on previously covered material (10 min).

-) No. 000 (a, b) p.50, No. 000

-) No. 000, No. 000, No. 000 (slide 11)

VII.Summing up the lesson (2 min).

-) How to compare fractions with the same denominators?

-) How to compare fractions with the same numerators?

-) How to compare fractions with different denominators and numerators?