Adding fractions with the same denominator. Adding and subtracting fractions and mixed numbers with different denominators

Open lesson

in mathematics grade 6b (remedial class VIII kind)

on the topic:

Adding Fractions

with the same denominators.

Type of lesson: learning new material.

Lesson type: lesson - fairy tale.

Class: 6.7 "B".

Goals:

Introduce students to the operations of adding and subtracting fractions with like denominators;

Tasks:

Correctional - educational:

Develop skills in adding fractions with like denominators;

Correctional - developmental:

Correct the development of logical and mathematical thinking while reciting the algorithm for adding fractions with like denominators and when doing written work in a notebook;

Correction of the development of students’ cognitive activity through completing tasks in non-standard situations;

Develop skills of attention and self-control.

Correctional and educational:

Instill interest in the subject based on connections with life and practice;

Formation of a mathematical culture of speech (correct pronunciation of fractions);

Develop self-esteem skills;

Lesson progress

Org. Moment.

1.Greeting

“Nice to see you guys. How are you feeling? Remember, if something seems difficult and doesn’t work out, then it’s not a problem, we will learn everything together!

2.getting ready to work

Guys, are you ready for the lesson?

I count on you, friends!

You are a good, friendly class,

Everything will work out for us!

Our lesson today is unusual; we will take you on a journey through a fairy tale we know and love.

There are many fairy tales in the world

Sad and funny.

And live in the world

We can't live without them!

Let the heroes of fairy tales

They give us warmth

May goodness forever

Evil wins!

Oral counting.

In the Thirtieth Kingdom lived the Tsar and his daughter Vasilisa the Wise, and in the Thirtieth Kingdom lived Ivan the Tsarevich. By the way, what number do you see on the board? Let me help you:

Anyone can a mile away

See fractional the line.

Above the line – numerator , know,

Below the line - denominator.

A fraction like that for sure

You have to call ordinary.

But the king did not want to give his Vasilisa to the first person he met. He decided on a task for Ivan that he could not cope with. And he says to Ivan: “Go there - I don’t know where, bring this, I don’t know what.” Ivan strained, grieved and went in search. But where to go, where to look?

Ivan, together with the Gray Wolf, set off on the road. They decided to first turn to Baba Yaga. And Baba Yaga prepared a task.

Oral calculation tasks. But guys, Ivan Tsarevich was not good at mathematics, should we help him?

State the numerator and denominator of the fraction

What does the numerator show and what does the denominator show? (The denominator shows how many shares are divided into, and the numerator shows how many such shares are taken.)

Comparison of fractions:

and 1 and and 1

And
5/5 and
And .

Well done, you completed the task. And now let’s follow the magic ball further, to the immortal Koshchei himself.

III. Updating basic knowledge.

You need to get to Koshchei through a labyrinth of fractional numbers.

Write these fractions on two lines: ,, , , , . Correct: , , .

Incorrect: , , .

Well done, you completed this task too.

So the magic ball brought Ivan and the Gray Wolf to Koshchei. And Koschey says: “I’m bored living here alone, but if you amuse me, then I’ll help. Complete my tasks."

1. Task No. 1 . Exercise.

Fizminutka :

The bear came out of the den.

He raised his legs once and twice.

He sat down and stood up. He sat down and stood up.

He put his paws behind his back.

Staggered, turned around

And he stretched a little.

1.Draw a circle of radiusr=2 cm.

2. Paint over

circle - yellow

circle - blue.

Write down which part of the circle is shaded and which part is not shaded.

Shaded- __________

Not painted over - _________

Think about how you can use action signs to make numbers And , get number . A ?

Task No. 2. Card No. 1 (Problem task).

So, what are we going to do in class today? Let’s write down in our notebooks the number and topic of the lesson “Adding and subtracting fractions with the same denominator.” Our goal is to learn how to add and subtract fractions with the same denominators. Let's look at an example:

Algorithm for adding fractions with like denominators : To add or subtract fractions with like denominators, add or subtract their numerators and leave the denominator the same.

VI. Formation of skills and abilities of students.

So the magic ball brought Ivan and the Gray Wolf to the Serpent Gorynych. He kept a box, and no one knew what was in it. But the Serpent Gorynych will not just give the box to Ivan. We need to help Ivan Tsarevich, and for this everyone needs to work independently, and the tasks for independent work are in the box (they go to the box and take the tasks). Card No. 2 (independent work). When you complete the tasks, you and I will check the answers and find out whether we helped Ivan Tsarevich or not.

Work in notebooks:homework : Solve a problem from another fairy tale.

Lesson summary. Grading.

So, the fairy tale ends here. Tell me, what did we do today? Let's repeat the rule again.

Today's lesson is over,

But everyone should know:

Knowledge, perseverance and work,
They will lead you to success in life!

VI . Reflection.

Guys, did you like the lesson? Choose the appropriate emoticon and stick it on the board. Thanks for the lesson. Goodbye

Fractions are ordinary numbers and can also be added and subtracted. But because they have a denominator, they require more complex rules than for integers.

Let's consider the simplest case, when there are two fractions with the same denominators. Then:

To add fractions with the same denominators, you need to add their numerators and leave the denominator unchanged.

To subtract fractions with the same denominators, you need to subtract the numerator of the second from the numerator of the first fraction, and again leave the denominator unchanged.

Within each expression, the denominators of the fractions are equal. By definition of adding and subtracting fractions we get:

As you can see, it’s nothing complicated: we just add or subtract the numerators and that’s it.

But even in such simple actions, people manage to make mistakes. What is most often forgotten is that the denominator does not change. For example, when adding them, they also begin to add up, and this is fundamentally wrong.

Getting rid of the bad habit of adding denominators is quite simple. Try the same thing when subtracting. As a result, the denominator will be zero, and the fraction will (suddenly!) lose its meaning.

Therefore, remember once and for all: when adding and subtracting, the denominator does not change!

Many people also make mistakes when adding several negative fractions. There is confusion with the signs: where to put a minus and where to put a plus.

This problem is also very easy to solve. It is enough to remember that the minus before the sign of a fraction can always be transferred to the numerator - and vice versa. And of course, don’t forget two simple rules:

1. Plus by minus gives minus;
2. Two negatives make an affirmative.

Let's look at all this with specific examples:

Task. Find the meaning of the expression:

In the first case everything is simple, but in the second we introduce minuses into the numerators of the fractions:

What to do if the denominators are different

You cannot add fractions with different denominators directly. At least, this method is unknown to me. However, the original fractions can always be rewritten so that the denominators become the same.

There are many ways to convert fractions. Three of them are discussed in the lesson “Reducing fractions to a common denominator”, so we will not dwell on them here. Let's look at some examples:

Task. Find the meaning of the expression:

In the first case, we reduce the fractions to a common denominator using the “criss-cross” method. In the second we will look for the NOC. Note that 6 = 2 · 3; 9 = 3 · 3. The last factors in these expansions are equal, and the first ones are relatively prime. Therefore, LCM(6, 9) = 2 3 3 = 18.

What to do if a fraction has an integer part

I can please you: different denominators in fractions are not the biggest evil. Much more errors occur when the whole part is highlighted in the addend fractions.

Of course, there are own addition and subtraction algorithms for such fractions, but they are quite complex and require a long study. Better use the simple diagram below:

1. Convert all fractions containing an integer part to improper ones. We obtain normal terms (even with different denominators), which are calculated according to the rules discussed above;
2. Actually, calculate the sum or difference of the resulting fractions. As a result, we will practically find the answer;
3. If this is all that was required in the problem, we perform the inverse transformation, i.e. We get rid of an improper fraction by highlighting the whole part.

The rules for moving to improper fractions and highlighting the whole part are described in detail in the lesson “What is a numerical fraction”. If you don’t remember, be sure to repeat it. Examples:

Task. Find the meaning of the expression:

Everything is simple here. The denominators inside each expression are equal, so all that remains is to convert all fractions to improper ones and count. We have:

To simplify the calculations, I have skipped some obvious steps in the last examples.

A small note on the last two examples, where fractions with the integer part highlighted are subtracted. The minus before the second fraction means that the entire fraction is subtracted, and not just its whole part.

Re-read this sentence again, look at the examples - and think about it. This is where beginners make a huge number of mistakes. They love to give such problems on tests. You will also encounter them several times in the tests for this lesson, which will be published shortly.

Summary: general calculation scheme

In conclusion, I will give a general algorithm that will help you find the sum or difference of two or more fractions:

1. If one or more fractions have an integer part, convert these fractions to improper ones;
2. Bring all the fractions to a common denominator in any way convenient for you (unless, of course, the writers of the problems did this);
3. Add or subtract the resulting numbers according to the rules for adding and subtracting fractions with like denominators;
4. If possible, shorten the result. If the fraction is incorrect, select the whole part.

Remember that it is better to highlight the whole part at the very end of the problem, immediately before writing down the answer.

Today we will talk about fractions. What horror this word inspires in many students, but in vain... Working with fractions is actually not that difficult. The main thing is to understand the rules. What are we going to do today?

Unfortunately, this topic is a weak link for many students, although it is one of the most basic in the study of mathematics.

So, let's figure it out. Let's start with why it is needed at all.

There are situations in our life when it is necessary to divide some whole object into a certain number of parts (in life - cut, saw, break off, etc.). Let's take pizza as an example:

Let's say you and your family ordered pizza (or baked it - as you like). There are four people in your family... You'll have to share)) And most likely you will try to divide the pizza into equal pieces so as not to offend anyone. In the end, each member of your family will get one piece of pizza (as will the rest of the family). And it is precisely in this case that the concept of a fraction will help us. The numerator of the fraction will indicate the part of the pizza you got, and the denominator will indicate the total number of parts (equal parts).

You can cut the pizza into 6 equal parts, or 7, or 12….

And now a little theory:

• any fraction consists of a numerator (the number written above the fraction sign) and a denominator (the number written below the fraction sign);
• the denominator shows how many parts the object is divided into, and the numerator shows how many of these parts are taken for some purpose.
• fraction shows attitude taken parts to the total number of parts of the object.

I suggest that you perform the suggested exercises (simulators) while studying (repeating) the topic. This will help consolidate knowledge and gain the skill of applying it in practice. It is recommended to work with the simulators in the order in which they are given in this article.

We have figured out the use of fractions in our lives. Now let's look at the types of fractions. Common fractions can be proper or improper...

Just don’t ooh and ahh)) It’s even simpler.

• correct a fraction is a fraction whose numerator is less than its denominator;
• wrong A fraction is a fraction whose numerator is greater than its denominator.

As I said above, fractions (now we are talking about fractions with the same denominators) can be compared. For this it is necessary to compare their numerators(the denominators are the same...)

Have you noticed that if the numerator and denominator are the same, then we get a whole object?))

Therefore, they say that if the numerator and denominator are equal, then the fraction is equal to one.

And one more important point: I hope you noticed))) the slash icon means the “division” action. And then it becomes completely clear that if a number is divided by itself, the result will be one. But here I’m getting ahead of myself and we’ll talk about this more in an article about reducing fractions...

Now let's look at adding and subtracting fractions with like denominators. The rule is very simple: to add (subtract) fractions with the same denominators, you need to add (subtract) their numerators, and leave the denominator the same.

And finally, let's test our knowledge with a test. You can pass this test only if you complete all tasks correctly. Only in this case can we say that the topic has been mastered. You can take the test an infinite number of times. And even if you passed the test 100% the first time, come back to this page in a few days and test your knowledge again. This will only strengthen your knowledge and develop your skill in working with such fractions.

P.S. But this is not all about fractions, because they are not only ordinary, but also decimal. And also occur in a mixed number (a number in which there is both an integer part and a fractional part)... But more on that in the following articles. Don't miss it.

Solving problems from the problem book Vilenkin, Zhokhov, Chesnokov, Shvartsburd for grade 5 on the topic:

1005 A salad was made from tomatoes weighing 5/16 kg and cucumbers weighing 9/16 kg. What is the mass of the salad?
SOLUTION

1006 The mass of the machine is 73/100 t, and the mass of its packaging is 23/100 t. Find the mass of the machine including the packaging.
SOLUTION

1007 On the first day, potatoes were planted on 2/7 of the plot, and on the second day on 3/7 of the plot. What part of the plot was planted with potatoes during these two days?
SOLUTION

1008 One brigade received 7/10 tons of nails, and the second 3/10 tons less. How many nails did the second brigade receive?
SOLUTION

1009 In two days, 10/11 fields were sown. On the first day, 4/11 fields were sown. What part of the field was sown on the second day?
SOLUTION

1010 The tank is 3/5 filled with gasoline, 1/5 of the tank has been poured into a barrel. What part of the tank remains filled with gasoline?
SOLUTION

1012 Find the value of the expression
SOLUTION

1013 Of the 11 greenhouses of the vegetable farm, 4 are planted with tomatoes, and 2 with cucumbers. What part of the greenhouses is occupied by cucumbers and tomatoes? Solve the problem in two ways.
SOLUTION

1014 An area of ​​300 hectares was allocated for forest planting. Spruce was planted on 3/10 of the plot, and pine on 4/10 of the plot. How many hectares are occupied by spruce and pine together?
SOLUTION

1015 The team decided to produce 175 items over plan. On the first day she produced 9/25 of this quantity, on the second day 13/25 of this quantity. How many products did the team produce in these two days? How many items does she have left to make?
SOLUTION

1016 11/17 fields of the vegetable farm were planted with potatoes. 1/17 more fields are sown with cucumbers than carrots, and 8/17 fields less than potatoes. What part of the field is sown with cucumbers and what part with carrots? What part of the field is occupied by potatoes, cucumbers and carrots together?
SOLUTION

1019 There were 2 quintals of 70 kg of fruit in the tent. Apples made up 5/9 of all fruits, and pears made up 1/9 of all fruits. How much is the mass of apples greater than the mass of pears? Solve the problem in two ways.
SOLUTION

1020 On the first day the tourist walked 5/14 of the entire route, and on the second day 7/14. It is known that during these two days the tourist walked 36 km. How many kilometers is the entire tourist route?
SOLUTION

1021 The first story took up 5/13 of the book, and the second story took up 2/13 of the book. It is known that the first story took up 12 pages more than the second. How many pages are in the whole book?
SOLUTION

1022 Using the equality 4/25 + 12/25= 16/25, find the values ​​of the expression and solve the equations
SOLUTION

1024 260 people go on an excursion. How many buses should be ordered if each bus should carry no more than 30 passengers?
SOLUTION

1025 Draw a line segment. Then draw a line segment whose length is equal to
SOLUTION

1026 Find the coordinates of points A, B, C, D, E, M, K (Fig. 128) and compare these coordinates with 1.
SOLUTION

1027 Calculate the perimeter and area of ​​triangle ABC (Fig. 129)
SOLUTION

1030 Find all values ​​of x for which the fraction x/15 is a regular fraction and the fraction 8/x is an improper fraction.
SOLUTION

1031 Name 3 proper fractions whose numerator is greater than 100. Name 3 improper fractions whose denominator is greater than 200.
SOLUTION

1033 The length of a rectangular parallelepiped is 8 m, width is 6 m and height is 12 m. Find the sum of the areas of the largest and smallest faces of this parallelepiped.
SOLUTION

1034 To produce 750 m of viscose fabric, 10 kg of cellulose is required. From 1 m3 of wood you can get 200 kg of cellulose. How many meters of viscose fabric can be obtained from 20 m3 of wood?
SOLUTION

1035 The combination lock has six buttons. To open it, you need to press the buttons in a certain sequence and enter a code. How many code options are there for this lock?
SOLUTION

1036 Solve the equation: a) (x - 111) · 59 = 11,918; b) 975(x - 615) = 12,675; c) (30,901 - a) : 605 = 51; d) 39,765: (b - 893) = 1205.
SOLUTION

1037 Solve the problem: 1) Out of 30 planted seeds, 23 germinated. What part of the planted seeds germinated? 2) 40 swans swam on the pond. Of these, 30 were white. What proportion of all swans were white swans?
SOLUTION

1038 Find the value of the expression: 1) 76 · (3569 + 2795) - (24,078 + 30,785); 2) (43 512-43 006) 805 - (48 987 + 297 305)
SOLUTION

1039 In the first hour, 5/17 of the entire road was cleared of snow, and in the second hour, 9/17 of the entire road. How much of the road was cleared of snow during these two hours? Which part of the road was cleared less in the first hour than in the second?
SOLUTION

1040 6/25 m of fabric was used for the dress for the first doll, and 9/25 m of fabric for the dress for the second doll. How much fabric did you use for both dresses? How much more fabric was used on the second doll's dress than on the first doll's dress?

Find the numerator and denominator. A fraction includes two numbers: the number that is located above the line is called the numerator, and the number that is located below the line is called the denominator. The denominator denotes the total number of parts into which a whole is divided, and the numerator is the number of such parts considered.

• For example, in the fraction ½ the numerator is 1 and the denominator is 2.

Determine the denominator. If two or more fractions have a common denominator, such fractions have the same number under the line, that is, in this case, a certain whole is divided into the same number of parts. Adding fractions with a common denominator is very easy, since the denominator of the summed fraction will be the same as the fractions being added. For example:

• The fractions 3/5 and 2/5 have a common denominator of 5.
• The fractions 3/8, 5/8, 17/8 have a common denominator of 8.