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Rule for comparing mixed fractions. Comparing mixed numbers




The purpose of the lesson: to form the skills of comparing mixed numbers.

Lesson objectives:

  1. Learn to compare mixed numbers.
  2. Develop thinking, attention.
  3. Cultivate accuracy while drawing rectangles.

Equipment: table "Ordinary fractions", a set of circles "Fractions and fractions"

During the classes

I. Organizational moment.

Write the date in a notebook.

What date is today? What month? what year? What month is it? What's the lesson?

II. oral work

1. Work on the plate:

347 999 200 127
  • Read numbers.
  • Name the largest and smallest number.
  • List the numbers in ascending or descending order.
  • Name the neighbors of each number.
  • Comparison of 1st and 2nd numbers.
  • Compare 2nd and 3rd number.
  • How much is 3 less than 4.
  • Expand the last number into the sum of bit terms, name: how many units are in this number, how many tens, how many hundreds.

2. What numbers are we studying now? (Fractional.)

  • Name fractional numbers (1 number each).
  • Name mixed numbers (1 number each)

3. Using the set on the magnets "Shares and fractions" show the numbers and.

Today we will learn how to compare such numbers. writing in the notebook of the topic of the lesson.

III. Studying the topic of the lesson.

1. Compare numbers using circles:

and

2. We build rectangles and mark the numbers and .

Conclusion: of two mixed numbers, the one with more integers is greater.

3. Work according to the textbook: p. 83, figure 12.

(Whole apples and slices shown.)

We read the rule in the textbook (teacher, then 2-3 times children)

IV. Fitness minute.

Conducted by the teacher and students for the muscles of the back and torso.


This article will talk about comparison of mixed numbers. First, we will figure out which mixed numbers are called equal and which are unequal. Next, we will give a rule for comparing unequal mixed numbers, which allows you to find out which number is greater and which is less, and consider examples. Finally, we will focus on comparing mixed numbers with natural numbers and common fractions.

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Equal and unequal mixed numbers

First you need to know which mixed numbers are called equal and which are unequal. Let us give the corresponding definitions.

Definition.

Equal Mixed Numbers are mixed numbers that have the same whole parts and fractional parts.

In other words, two mixed numbers are said to be equal if their entries are exactly the same. If the entries of mixed numbers differ, then such mixed numbers are called unequal.

Definition.

Unequal mixed numbers are mixed numbers whose entries are different.

Voiced definitions allow you to determine at a glance whether given mixed numbers are equal or not. For example, mixed numbers and equal, since their entries are exactly the same. These numbers have equal integer parts and equal fractional parts. And mixed numbers and are unequal, since they have unequal integer parts. Other examples of unequal mixed numbers are and , as well as and .

Sometimes it becomes necessary to find out which of two unequal mixed numbers is greater than the other, and which is less. How this is done, we will consider in the next paragraph.

Comparing mixed numbers

Comparing mixed numbers can be reduced to comparing ordinary fractions. To do this, it is enough to convert mixed numbers to improper fractions.

For example, let's compare a mixed number and a mixed number by representing them as improper fractions. We have and . So the comparison of the original mixed numbers is reduced to the comparison of fractions with different denominators and . Since , then .

Comparing mixed numbers by comparing their equal fractions is not the best solution. It is much more convenient to use the following mixed number comparison rule: more is the mixed number, the integer part of which is greater, but if the integer parts are equal, then the greater is the mixed number, the fractional part of which is greater.

Consider how the comparison of mixed numbers occurs according to the voiced rule. To do this, we will analyze the solutions of examples.

Example.

Which of the mixed numbers and more?

Solution.

The integer parts of the compared mixed numbers are equal, so the comparison is reduced to comparing the fractional parts and . Since then . So the mixed number is greater than the mixed number.

Answer:

Comparing a mixed number and a natural number

Let's figure out how to compare a mixed number and a natural number.

Fair enough rule for comparing a mixed number with a natural number: if the integer part of the mixed number is less than the given natural number, then the mixed number is less than the given natural number, and if the integer part of the mixed number is greater than or equal to the given mixed number, then the mixed number is greater than the given natural number.

Let's look at examples of comparing a mixed number and a natural number.

Example.

Compare numbers 6 and .

Solution.

The integer part of the mixed number is 9 . Since it is greater than the natural number 6, then .

Answer:

Example.

Given a mixed number and a natural number 34, which number is smaller?

Solution.

The integer part of the mixed number is less than 34 (11<34 ), поэтому .

Answer:

The mixed number is less than the number 34 .

Example.

Perform a comparison between the number 5 and the mixed number.

Solution.

The integer part of this mixed number is equal to the natural number 5 , therefore, this mixed number is greater than 5 .

Answer:

To conclude this subsection, we note that any mixed number is greater than one. This statement follows from the rule for comparing a mixed number and a natural number, and also from the fact that the integer part of any mixed number is either greater than 1 or equal to 1.

Comparing a mixed number and a common fraction

First, let's talk about comparing a mixed number and a proper fraction. Any proper fraction is less than 1 (see proper and improper fractions), so any proper fraction is less than any mixed number (since any mixed number is greater than 1).

We continue to study fractions. Today we will talk about their comparison. The topic is interesting and useful. It will allow the beginner to feel like a scientist in a white coat.

The essence of comparing fractions is to find out which of the two fractions is greater or less.

To answer the question which of the two fractions is greater or less, use such as more (>) or less (<).

Mathematicians have already taken care of ready-made rules that allow you to immediately answer the question of which fraction is larger and which is smaller. These rules can be safely applied.

We will look at all these rules and try to figure out why this happens.

Lesson content

Comparing fractions with the same denominators

The fractions to be compared come across different. The most successful case is when fractions have the same denominators, but different numerators. In this case, the following rule applies:

Of two fractions with the same denominator, the larger fraction is the one with the larger numerator. And accordingly, the smaller fraction will be, in which the numerator is smaller.

For example, let's compare fractions and and answer which of these fractions is greater. Here the denominators are the same, but the numerators are different. A fraction has a larger numerator than a fraction. So the fraction is greater than . So we answer. Reply using the more icon (>)

This example can be easily understood if we think about pizzas that are divided into four parts. more pizzas than pizzas:

Comparing fractions with the same numerator

The next case we can get into is when the numerators of the fractions are the same, but the denominators are different. For such cases, the following rule is provided:

Of two fractions with the same numerator, the fraction with the smaller denominator is larger. The fraction with the larger denominator is therefore smaller.

For example, let's compare fractions and . These fractions have the same numerator. A fraction has a smaller denominator than a fraction. So the fraction is greater than the fraction. So we answer:

This example can be easily understood if we think about pizzas that are divided into three and four parts. more pizzas than pizzas:

Everyone will agree that the first pizza is bigger than the second one.

Comparing fractions with different numerators and different denominators

It often happens that you have to compare fractions with different numerators and different denominators.

For example, compare fractions and . To answer the question which of these fractions is greater or less, you need to bring them to the same (common) denominator. Then it will be easy to determine which fraction is greater or less.

Let's bring the fractions to the same (common) denominator. Find (LCM) the denominators of both fractions. The LCM of the denominators of the fractions and that number is 6.

Now we find additional factors for each fraction. Divide the LCM by the denominator of the first fraction. LCM is the number 6, and the denominator of the first fraction is the number 2. Divide 6 by 2, we get an additional factor of 3. We write it over the first fraction:

Now let's find the second additional factor. Divide the LCM by the denominator of the second fraction. LCM is the number 6, and the denominator of the second fraction is the number 3. Divide 6 by 3, we get an additional factor of 2. We write it over the second fraction:

Multiply the fractions by their additional factors:

We came to the conclusion that fractions that had different denominators turned into fractions that had the same denominators. And we already know how to compare such fractions. Of two fractions with the same denominators, the larger fraction is the one with the larger numerator:

The rule is the rule, and we will try to figure out why more than . To do this, select the integer part in the fraction. There is no need to select anything in the fraction, since this fraction is already regular.

After selecting the integer part in the fraction, we get the following expression:

Now you can easily understand why more than . Let's draw these fractions in the form of pizzas:

2 whole pizzas and pizzas, more than pizzas.

Subtraction of mixed numbers. Difficult cases.

When subtracting mixed numbers, sometimes you find that things don't go as smoothly as you'd like. It often happens that when solving an example, the answer is not what it should be.

When subtracting numbers, the minuend must be greater than the subtrahend. Only in this case will a normal response be received.

For example, 10−8=2

10 - reduced

8 - subtracted

2 - difference

The minus 10 is greater than the subtracted 8, so we got the normal answer 2.

Now let's see what happens if the minuend is less than the subtrahend. Example 5−7=−2

5 - reduced

7 - subtracted

−2 is the difference

In this case, we go beyond the numbers we are used to and find ourselves in the world of negative numbers, where it is too early for us to walk, and even dangerous. To work with negative numbers, you need the appropriate mathematical background, which we have not received yet.

If, when solving examples for subtraction, you find that the minuend is less than the subtrahend, then you can skip such an example for now. It is permissible to work with negative numbers only after studying them.

The situation is the same with fractions. The minuend must be greater than the subtrahend. Only in this case it will be possible to get a normal answer. And in order to understand whether the reduced fraction is greater than the subtracted one, you need to be able to compare these fractions.

For example, let's solve an example.

This is a subtraction example. To solve it, you need to check whether the reduced fraction is greater than the subtracted one. more than

so we can safely return to the example and solve it:

Now let's solve this example

Check if the reduced fraction is greater than the subtracted one. We find that it is less:

In this case, it is more reasonable to stop and not continue further calculation. We will return to this example when we study negative numbers.

It is also desirable to check mixed numbers before subtracting. For example, let's find the value of the expression .

First, check whether the reduced mixed number is greater than the subtracted one. To do this, we translate mixed numbers into improper fractions:

We got fractions with different numerators and different denominators. To compare such fractions, you need to bring them to the same (common) denominator. We will not describe in detail how to do this. If you're having trouble, be sure to repeat.

After reducing the fractions to the same denominator, we get the following expression:

Now we need to compare fractions and . These are fractions with the same denominators. Of two fractions with the same denominator, the larger fraction is the one with the larger numerator.

A fraction has a larger numerator than a fraction. So the fraction is greater than the fraction.

This means that the minuend is greater than the subtrahend.

So we can go back to our example and boldly solve it:

Example 3 Find the value of an expression

Check if the minuend is greater than the subtrahend.

Convert mixed numbers to improper fractions:

We got fractions with different numerators and different denominators. We bring these fractions to the same (common) denominator:

Now let's compare the fractions and . A fraction has a numerator less than a fraction, so the fraction is smaller than the fraction

Outline plan math lesson in 6th grade

Lesson topic: "Comparison of Mixed Numbers"

The purpose of the lesson: learn the rules for comparing mixed numbers; to consolidate the skills and abilities of comparing ordinary fractions and mixed numbers when solving problems.

Tasks:

    to generalize students' knowledge about ordinary fractions and mixed numbers, to form the ability to compare ordinary fractions and mixed numbers;

    continue work on the development of logical thinking, memory, imagination, the formation of mathematically literate speech;

    to instill in students a sense of responsibility, to improve the skills of independent activity.

Lesson type: lesson learning new knowledge.

Equipment: projector, interactive whiteboard, handouts.

Lesson structure:

1. Organizational moment (3 min).

2. Actualization of knowledge (10 min).

3. Learning new material (8 min).

4. Physical education (1 min).

5. Consolidation of the passed (15min).

6. Homework (1 min).

7. Summary of the lesson (2 min).

During the classes.

I. Organizing time . (Slide #2)

Guys, open notebooks, write down the date and topic of the lesson “Comparison of mixed numbers”.

Today we will study a new topic, learn how to compare mixed numbers. But before that, we must repeat one important theme. And what, you will know ifsolve the puzzle :

( fraction )

II. Knowledge update. oral work .

1) - Look at the screen (slide number 3 ).

- What part of the figure is shaded? write down a fraction (3/8)

What is the name of the number below the line? (denominator )

What does the denominator of a fraction show? (The denominator shows how many equal parts the whole is divided into. )

What is the name of the number above the line? (numerator )

What does the numerator of a fraction show? (the numerator shows how many parts were taken )

2) - The next task "Find the extra "(slide number 4) :

A) numerator sum; denominator; fraction.

B) ;. ()

Why is it redundant? (this is an improper fraction, the rest are correct )

What fractions are right? (proper fractions have numerator less than denominator)

- What fractions are called improper? (For improper fractions, the numerator is greater than or equal to the denominator)

AT) ;. ()

Why is it redundant? (it's a mixed number) I write on the board

What are the parts of a mixed number? (from a whole number and a fraction or an integer part and a fractional part )

3) Independent work on cards.

Now let's remember how ordinary fractions are compared. To do this, we will executeindependent work . We write down the solutions on the sheets with tasks:

…. ; …. ;

…. ; …. ;

…. ; …. .

Let's check your solutions. Whoever has it right, without mistakes - put "5", who has 1-2 mistakes - "4", who has 3 or more - "3".

Self-examination (on slide number 5 answers)

What are the rules for comparing common fractions?(with rules for comparing ordinary fractions with the same denominators and the same numerators)

Let's read the rules of comparison aloud together:

Rule 1: (Slide #6)

Of two fractions with the same denominators, the greater is the fraction with numerator is greater .

Rule 2: (Slide #6)

Of two fractions with the same numerator, the greater is the fraction with denominator less .

    Exploring a New Topic Comparing mixed numbers »

When comparing mixed numbers, there can be two cases of comparison.

Let's consider the first case. Look at the screenSlide number 7 ).

What mixed numbers are shown on the screen? (and )

Write them down in your notebook:

Name the integer part of each number. (3 and 2)

Are the whole parts the same or different? (various )

Which mixed number has the largest integer part? (In the first )

Which number is greater? ()

- What can we conclude? Continue …

Meansto compare mixed numbers, first compare the whole parts.

Conclusion : Of two mixed numbers, the greater is the one in which whole part…..more .

Examples for consolidation (Slide number 8)

- Do the following verbally:

Read and compare the numbers: and; and; and. That more?

Continued and learning a new topic

Let's consider the second case. What mixed numbers are shown on the next slide?(Slide number 9)

Write mixed numbers in your notebook

What can be said about the integer parts of these mixed numbers? (they are identical )

How do you think how to compare two mixed numbers with the same integer parts? (look at fractional parts or fractions )

Which is larger ¾ or ¼? (¾)

Which number is greater? ()

- So if the integer parts are the same, then look at the fractional parts

AT conclusion: (Slide number 8) Continue …

Of two mixed numbers with the same integer parts, the larger number is whom fractional part……more .

    Physical education (slide number 9).

Once - got up, stretched.

Two - bent down, unbent.

Three - in the hands of three claps,

Three head nods.

Four - arms wider.

Five - wave your hands.

Six - sit quietly at the desk.

v. Consolidation of the studied .

1 ) Work with the textbook .

Opening textbooksPage 84 decide № 317 (2)

Goes to the board ... .., and the rest decide in notebooks.

2) - Solve the problem orally (on Slide #10) .

Masha has an orange, Alena has an orange, Olya has an orange. Who has more orange? Who has less orange?

3) Game "Math beads".

Beads are drawn on the board. You need to take turns going to the board, come up with and write in circlesmixed numbers in ascending order .

VI. Lesson summary .

What topic did you study in class today?

How to compare mixed numbers with different integer parts?

How to compare mixed numbers with the same integer parts?

- Grades per lesson : .

Thank you for your work!

VI I . Homework : No. 320 p. 85. (compare mixed)

Additional task for independent work (at the end of the lesson):

Option 1.

Compare numbers:

…. ; … ; 10 ….. 10

…. ; … ; ….. 3

Independent work (for 3 minutes)

Option 1

…. ; …. ;

…. ; …. ;

…. ; …. .

The purpose of the lesson: to form the skills of comparing mixed numbers.

Lesson objectives:

  1. Learn to compare mixed numbers.
  2. Develop thinking, attention.
  3. Cultivate accuracy while drawing rectangles.

Equipment: table "Ordinary fractions", a set of circles "Fractions and fractions"

During the classes

I. Organizational moment.

Write the date in a notebook.

What date is today? What month? what year? What month is it? What's the lesson?

II. oral work

1. Work on the plate:

347 999 200 127
  • Read numbers.
  • Name the largest and smallest number.
  • List the numbers in ascending or descending order.
  • Name the neighbors of each number.
  • Comparison of 1st and 2nd numbers.
  • Compare 2nd and 3rd number.
  • How much is 3 less than 4.
  • Expand the last number into the sum of bit terms, name: how many units are in this number, how many tens, how many hundreds.

2. What numbers are we studying now? (Fractional.)

  • Name fractional numbers (1 number each).
  • Name mixed numbers (1 number each)

3. Using the set on the magnets "Shares and fractions" show the numbers and.

Today we will learn how to compare such numbers. writing in the notebook of the topic of the lesson.

III. Studying the topic of the lesson.

1. Compare numbers using circles:

and

2. We build rectangles and mark the numbers and .

Conclusion: of two mixed numbers, the one with more integers is greater.

3. Work according to the textbook: p. 83, figure 12.

(Whole apples and slices shown.)

We read the rule in the textbook (teacher, then 2-3 times children)

IV. Fitness minute.

Conducted by the teacher and students for the muscles of the back and torso.

V. Fixing the material.

1. Repetition according to the table "Ordinary fractions".

(Numbers where the integer parts are the same are covered in the next lesson.)

2. Compare.

VI. Homework on individual cards, learn the rule on page 83 of the textbook.

VII. Individual work on cards.

VIII. Summary of the lesson.

Grading.