The purpose of the lesson: develop skills in comparing mixed numbers.

Lesson objectives:

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

Equipment: table “Ordinary fractions”, 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 is the lesson?

II. Oral work

1. Work according to the plate:

347

999

200

127

• Read the numbers.
• Name the largest and smallest number.
• Name the numbers in descending and ascending order.
• Name the neighbors of each number.
• Comparison of 1st and 2nd numbers.
• Compare numbers 2 and 3.
• How much is 3 less than 4?
• Decompose the last number into the sum of the digit terms, name: how many units are there in this number, how many tens are there, how many hundreds are there.

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

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

3. Using the “Shares and Fractions” magnet set, show the numbers and .

Today we will learn to compare such numbers. write down the topic of the lesson in your notebook.

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 number with more integers is greater.

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

(Whole apples and lobes are depicted.)

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

IV. Physical education moment.

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

The purpose of the lesson: develop skills in comparing mixed numbers.

Lesson objectives:

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

Equipment: table “Ordinary fractions”, 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 is the lesson?

II. Oral work

1. Work according to the plate:

347

999

200

127

• Read the numbers.
• Name the largest and smallest number.
• Name the numbers in descending and ascending order.
• Name the neighbors of each number.
• Comparison of 1st and 2nd numbers.
• Compare numbers 2 and 3.
• How much is 3 less than 4?
• Decompose the last number into the sum of the digit terms, name: how many units are there in this number, how many tens are there, how many hundreds are there.

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

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

3. Using the “Shares and Fractions” magnet set, show the numbers and .

Today we will learn to compare such numbers. write down the topic of the lesson in your notebook.

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 number with more integers is greater.

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

(Whole apples and lobes are depicted.)

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

IV. Physical education moment.

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 whole parts are the same are covered in the next lesson.)

2. Compare.

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

VII. Individual work by cards.

VIII. Lesson summary.

Grading.

Comparison rules ordinary fractions depend on the type of fraction (proper, improper, mixed fraction) and on the denominator (same or different) of the fractions being compared.

This section discusses options for comparing fractions that have the same numerators or denominators.

Rule. To compare two fractions with the same denominators, you need to compare their numerators. Greater (less) is a fraction whose numerator is greater (less).

For example, compare fractions:

Rule. To compare proper fractions with like numerators, you need to compare their denominators. Greater (less) is a fraction whose denominator is less (greater).

For example, compare fractions:

Comparing proper, improper and mixed fractions with each other

Rule. Improper and mixed fractions are always larger than any proper fraction.

A proper fraction is by definition less than 1, so improper and mixed fractions (those containing a number equal to or greater than 1) are greater than a proper fraction.

Rule. Of two mixed fractions, the greater (smaller) is the one with whole part fractions are greater (less). When the whole parts of mixed fractions are equal, the fraction with the larger (smaller) fractional part is greater (smaller).

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

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

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

Mathematicians have already taken care of ready-made rules that allow them 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 that need to be compared are different. The best case is when the fractions have the same denominators, but different numerators. In this case, the following rule applies:

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

For example, let's compare fractions and answer which of these fractions is larger. Here the denominators are the same, but the numerators are different. The fraction has a greater numerator than the fraction. This means the fraction is greater than . That's how we answer. You must answer using the more icon (>)

This example can be easily understood if we remember about pizzas, which are divided into four parts. There are more pizzas than pizzas:

Comparing fractions with the same numerators

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 numerators, the fraction with the smaller denominator is greater. And accordingly, the fraction whose denominator is larger is smaller.

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

This example can be easily understood if we remember about pizzas, which are divided into three and four parts. There are more pizzas than pizzas:

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

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 you can easily determine which fraction is greater or less.

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

Now we find additional factors for each fraction. Let's 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 above the first fraction:

Now let's find the second additional factor. Let's 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 above the second fraction:

Let's 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 denominator, the fraction with the larger numerator is greater:

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

After isolating the integer part in the fraction, we obtain the following expression:

Now you can easily understand why more than . Let's draw these fractions as pizzas:

2 whole pizzas and pizzas, more than pizzas.

Subtraction of mixed numbers. Difficult cases.

When subtracting mixed numbers, you can sometimes find that things aren't going 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 answer be received.

For example, 10−8=2

10 - decrementable

8 - subtrahend

2 - difference

The minuend 10 is greater than the subtrahend 8, so we get the normal answer 2.

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

5—decreasable

7 - subtrahend

−2 — difference

In this case, we go beyond the limits of the numbers we are accustomed 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, we need appropriate mathematical training, which we have not yet received.

If, when solving subtraction examples, 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 will it be possible to get a normal answer. And to understand whether the fraction being reduced is greater than the fraction being subtracted, you need to be able to compare these fractions.

For example, let's solve the example.

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

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

Now let's solve this example

We check whether the fraction being reduced is greater than the fraction being subtracted. We find that it is less:

In this case, it is wiser to stop and not continue further calculation. Let's return to this example when we study negative numbers.

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

First, let's check whether the mixed number being reduced is greater than the mixed number being subtracted. To do this, we convert mixed numbers to improper fractions:

We received 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 have difficulty, be sure to repeat.

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

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

The fraction has a greater numerator than the fraction. This means that the fraction is greater than the fraction.

This means that the minuend is greater than the subtrahend

This means we can return to our example and safely solve it:

Example 3. Find the value of an expression

Let's check whether the minuend is greater than the subtrahend.

Let's convert mixed numbers to improper fractions:

We received fractions with different numerators and different denominators. Let us reduce these fractions to the same (common) denominator:

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