Multiplication of numbers with different signs (6th grade). Multiplying fractions with different signs

Lesson objectives:

Educational:

formulating rules for multiplying numbers with the same and different signs;
mastering and improving the skills of multiplying numbers with different signs.

Educational:

development of mental operations: comparison, generalization, analysis, analogy;
skills development independent work;
broadening the horizons of students.

Educational:

fostering a record-keeping culture;
education of responsibility, attention;
nurturing interest in the subject.

Lesson type: learning new material.

Equipment: computer, multimedia projector, cards for the game “Mathematical Combat”, tests, knowledge cards.

Posters on the walls:

Knowledge is the most excellent of possessions. Everyone strives for it, but it does not come on its own.
Al-Biruni
In everything I want to get to the very essence...
B. Pasternak

Lesson Plan

Organizational moment (1 min).
Introductory speech by the teacher (3 min).
Oral work (10 min).
Presentation of the material (15 min).
Mathematical chain (5 min).
Homework(2 min).
Test (6 min).
Lesson summary (3 min).

Lesson progress

I. Organizational moment

students' readiness for the lesson.

II. Teacher's opening speech

Guys, we met with you today not in vain, but for fruitful work: gaining knowledge.

Since the universe has existed,
There is no one who does not need knowledge.
Whatever language and age we choose,
Man has always strived for knowledge...
Rudaki

In class we will study new material, consolidate it, work independently, evaluate yourself and your comrades. Everyone has a knowledge card on their desk, in which our lesson is divided into stages. The points you earned on different stages you yourself will enter the lesson into this map. And at the end of the lesson we will summarize. Place these cards in a visible place.

III. Oral work (in the form of the game “Mathematical Combat”)

Guys, before we start new topic, let's repeat what we learned earlier. Everyone has a sheet of paper with the game “Mathematical Combat” on their desk. The vertical and horizontal columns contain the numbers that need to be added. These numbers are marked with dots. We will write the answers in those cells on the field where the dots are.

Three minutes to complete. We started work.

Now we exchanged works with our desk neighbor and check them with each other. If you think that the answer is incorrect, then carefully cross it out and write the correct one next to it. Let's check.

Now let’s check the answers with the screen ( The correct answers are projected on the screen).

For correctly solved

5 tasks are given 5 points;
4 tasks – 4 points;
3 tasks – 3 points;
2 tasks – 2 points;
1 task – 1 point.

Well done. They put everything aside. Guys, let’s enter the number of points scored for the “Mathematical Battle” into our knowledge cards ( Appendix 1).

IV. Presentation of the material

Open the workbooks. Write down the number, great job.

What operations on positive and negative numbers do you know?
How to add two negative numbers?
How to add two numbers with different signs?
How to subtract numbers with different signs?
You always use the word "module". What is the modulus of a number? A?

Today's lesson topic is also related to the operation of numbers of different signs. But it was hidden in an anagram, in which you need to swap letters and get a familiar word. Let's try to figure it out.

ENOZHEUMNI

We write down the topic of the lesson: “Multiplication.”

The purpose of our lesson: to get acquainted with the multiplication of positive and negative numbers and formulate rules for multiplying numbers with both the same and different signs.

All attention to the board. Before you is a table with problems, solving which we will formulate the rules for multiplying positive and negative numbers.

2*3 = 6°C;
–2*3 = –6°С;
–2*(–3) = 6°С;
2*(–3) = –6°С;

1. The air temperature rises by 2°C every hour. Now the thermometer shows 0°C ( Appendix 2– Thermometer) (slide 1 on the computer).

How much did you receive?(6 ° WITH).
Someone will write the solution on the board, and we are all in notebooks.
Let's look at the thermometer, did we get the correct answer? (slide 2 on the computer).

2. The air temperature drops by 2°C every hour. The thermometer now shows 0°C (slide 3 on the computer). What air temperature will the thermometer show after 3 hours?

How much did you receive?(–6 ° WITH).
We write down the corresponding solution on the board and in notebooks. Analogy with task 1.
.(slide 4 on the computer).

3. The air temperature drops by 2°C every hour. The thermometer now shows 0°C (slide 5 on the computer).

How much did you receive?(6 ° WITH).
We write down the corresponding solution on the board and in notebooks. Analogy with tasks 1 and 2.
Let's compare the result with the thermometer reading.(slide 6 on the computer).

4. The air temperature rises by 2°C every hour. The thermometer now shows 0°C (slide 7 on the computer). What air temperature did the thermometer show 3 hours ago?

How much did you receive?(–6 ° WITH).
We write down the corresponding solution on the board and in notebooks. Analogy with tasks 1-3.
Let's compare the result with the thermometer reading.(slide 8 on the computer).

Look at your results. When multiplying numbers with the same signs (examples 1 and 3), what sign did you get the answer? (positive).

Fine. But in example 3, both factors are negative, and the answer is positive. What mathematical concept allows you to move from negative numbers to positive ones? (module).

Attention rule: To multiply two numbers with the same signs, you need to multiply their absolute values and put a plus sign in front of the result. (2 people repeat).

Let's return to example 3. What are the modules (–2) and (–3) equal to? Let's multiply these modules. How much did you receive? With what sign?

When multiplying numbers with different signs (examples 2 and 4), what sign did you get the answer? (negative).

Formulate your own rule for multiplying numbers with different signs.

Rule: When multiplying numbers with different signs, you need to multiply their modules and put a minus sign in front of the result. (2 people repeat).

Let's return to examples No. 2 and No. 4. What are the magnitudes of their factors? Let's multiply these modules. How much did you receive? What sign should be given as a result?

Using these two rules, you can also multiply fractions: decimal, mixed, ordinary.

There are several examples on the board in front of you. We will decide three together with me, and the rest on our own. Pay attention to the recording and design.

Well done. Let's open the textbooks and mark the rules that need to be learned for the next lesson (page 190, §7 (point 35)). Knowing these rules will help you quickly master the division of positive and negative numbers in the future.

V. Mathematical chain

And now Dunno wants to check how you have learned the new material and will ask you a few questions. We must write down the solution and answers in notebooks ( Appendix 3– Mathematical chain).

Computer presentation
Hello guys. I see you are very smart and inquisitive, so I want to ask you a few questions. Be careful, especially with signs.
My first question is: multiply (–3) by (–13).
Second question: multiply what you got in the first task by (–0,1).
Third question: multiply the result of the second task by (–2).
Fourth question: multiply (-1/3) by the result of the third task.
And the last, fifth question: calculate the freezing point of mercury by multiplying the result of the fourth task by 15.
Thanks for the work. I wish you success.

Guys, let's check how we completed the tasks. Everyone got up.

How much did you get in the first task?

Those who have a different answer, sit down, and those who sit down, we give ourselves 0 points for the mathematical chain on the knowledge record card. The rest don't put anything.

How much did you get in the second task?

If you have a different answer, sit down and add 1 point to your knowledge card for the mathematical chain.

How much did you get in the third task?

If you have a different answer, sit down and add 2 points to your knowledge card for the mathematical chain.

How much did you get in the fourth task?

For those who have a different answer, sit down and add 3 points to your knowledge record card for the mathematical chain.

How much did you get in the fifth task?

For those who have a different answer, sit down and add 4 points to your knowledge record card for the mathematical chain. The remaining guys solved all 5 tasks correctly. Sit down, you give yourself 5 points for the mathematical chain on your knowledge record card.

What is the freezing point of mercury?(–39 °C).

VI. Homework

§7 (clause 35, page 190), No. 1121 – textbook: Mathematics. 6th grade: [N.Ya.Vilenkin and others]

Creative task: Write a problem on multiplying positive and negative numbers.

VII. Test

Let's move on to the next stage of the lesson: performing the test ( Appendix 4).

You need to solve the tasks and circle the number of the correct answer. For the first two correctly completed tasks you will receive 1 point, for the 3rd task - 2 points, for the 4th task - 3 points. We started work.

Δ –1 point;
o –2 points;
–3 points.

Now let’s write down the numbers of the correct answers in the table below the test. Let's check the results. You should get the number 1418 in the empty cells (I write on the board). Whoever received it puts 7 points on the knowledge card. Those who made mistakes put the number of points scored only for correctly completed tasks on the knowledge record card.

The Great Great War lasted exactly 1418 days. Patriotic War, a victory in which the Russian people came at a heavy price. And on May 9, 2010 we will celebrate the 65th anniversary of the Victory over Nazi Germany.

VIII. Lesson summary

Now let's count total quantity the points you scored for the lesson, and the results will be entered into the students’ knowledge record card. Then we deal these cards.

15 – 17 points – score “5”;
10 – 14 points – score “4”;
less than 10 points – score “3”.

Raise your hands who received “5”, “4”, “3”.

What topic did we cover today?
How to multiply numbers with the same signs; with different signs?

So, our lesson has come to an end. I want to say THANK YOU for your work in this lesson.

) and denominator by denominator (we get the denominator of the product).

Formula for multiplying fractions:

For example:

Before you begin multiplying numerators and denominators, you need to check whether the fraction can be reduced. If you can reduce the fraction, it will be easier for you to make further calculations.

Dividing a common fraction by a fraction.

Dividing fractions involving natural numbers.

It's not as scary as it seems. As in the case of addition, we convert the integer into a fraction with one in the denominator. For example:

Multiplying mixed fractions.

Rules for multiplying fractions (mixed):

convert mixed fractions to improper fractions;
multiplying the numerators and denominators of fractions;
reduce the fraction;
If you get an improper fraction, then we convert the improper fraction into a mixed fraction.

Pay attention! To multiply a mixed fraction by another mixed fraction, you first need to convert them to the form of improper fractions, and then multiply according to the rule for multiplying ordinary fractions.

The second way to multiply a fraction by a natural number.

It may be more convenient to use the second method of multiplying a common fraction by a number.

Pay attention! To multiply a fraction by natural number It is necessary to divide the denominator of the fraction by this number, and leave the numerator unchanged.

From the example given above, it is clear that this option is more convenient to use when the denominator of a fraction is divided without a remainder by a natural number.

Multistory fractions.

In high school, three-story (or more) fractions are often encountered. Example:

To bring such a fraction to its usual form, use division through 2 points:

Pay attention! When dividing fractions, the order of division is very important. Be careful, it's easy to get confused here.

Please note For example:

When dividing one by any fraction, the result will be the same fraction, only inverted:

Practical tips for multiplying and dividing fractions:

1. The most important thing when working with fractional expressions is accuracy and attentiveness. Do all calculations carefully and accurately, concentratedly and clearly. It's better to write a few extra lines in your draft than to get lost in mental calculations.

2. In tasks with different types fractions - go to the form of ordinary fractions.

3. We reduce all fractions until it is no longer possible to reduce.

4. Multi-storey fractional expressions we bring them into ordinary form, using division through 2 points.

5. Divide a unit by a fraction in your head, simply turning the fraction over.

Ordinary fractional numbers first meet schoolchildren in the 5th grade and accompany them throughout their lives, since in everyday life it is often necessary to consider or use an object not as a whole, but in separate pieces. Start studying this topic - shares. Shares are equal parts, into which this or that object is divided. After all, it is not always possible to express, for example, the length or price of a product as a whole number; parts or fractions of some measure should be taken into account. Formed from the verb “to split” - to divide into parts, and having Arabic roots, the word “fraction” itself arose in the Russian language in the 8th century.

Fractional expressions have long been considered the most difficult branch of mathematics. In the 17th century, when first textbooks on mathematics appeared, they were called “broken numbers,” which was very difficult for people to understand.

Modern look simple fractional remainders, the parts of which are separated by a horizontal line, were first promoted by Fibonacci - Leonardo of Pisa. His works are dated to 1202. But the purpose of this article is to simply and clearly explain to the reader how mixed fractions are multiplied with different denominators.

Multiplying fractions with different denominators

Initially it is worth determining types of fractions:

correct;
incorrect;
mixed.

Next, you need to remember how fractional numbers are multiplied with same denominators. The very rule of this process is not difficult to formulate independently: the result of multiplying simple fractions with identical denominators is a fractional expression, the numerator of which is the product of the numerators, and the denominator is the product of the denominators of these fractions. That is, in fact, the new denominator is the square of one of the initially existing ones.

When multiplying simple fractions with different denominators for two or more factors the rule does not change:

a/b * c/d = a*c / b*d.

The only difference is that the resulting number under the fractional line will be the product of different numbers and, naturally, the square of one numerical expression it is impossible to name it.

It is worth considering the multiplication of fractions with different denominators using examples:

8/ 9 * 6/ 7 = 8*6 / 9*7 = 48/ 63 = 16/2 1 ;
4/ 6 * 3/ 7 = 2/ 3 * 3/7 <> 2*3 / 3*7 = 6/ 21 .

The examples use methods for reducing fractional expressions. You can only reduce numerator numbers with denominator numbers; adjacent factors above or below the fraction line cannot be reduced.

Along with simple fractions, there is the concept of mixed fractions. A mixed number consists of an integer and a fractional part, that is, it is the sum of these numbers:

1 4/ 11 =1 + 4/ 11.

How does multiplication work?

Several examples are provided for consideration.

2 1/ 2 * 7 3/ 5 = 2 + 1/ 2 * 7 + 3/ 5 = 2*7 + 2* 3/ 5 + 1/ 2 * 7 + 1/ 2 * 3/ 5 = 14 + 6/5 + 7/ 2 + 3/ 10 = 14 + 12/ 10 + 35/ 10 + 3/ 10 = 14 + 50/ 10 = 14 + 5=19.

The example uses multiplication of a number by ordinary fractional part, the rule for this action can be written as:

a* b/c = a*b /c.

In fact, such a product is the sum of identical fractional remainders, and the number of terms indicates this natural number. Special case:

4 * 12/ 15 = 12/ 15 + 12/ 15 + 12/ 15 + 12/ 15 = 48/ 15 = 3 1/ 5.

There is another solution to multiplying a number by a fractional remainder. You just need to divide the denominator by this number:

d* e/f = e/f: d.

This technique is useful to use when the denominator is divided by a natural number without a remainder or, as they say, by a whole number.

Convert mixed numbers to improper fractions and obtain the product in the previously described way:

1 2/ 3 * 4 1/ 5 = 5/ 3 * 21/ 5 = 5*21 / 3*5 =7.

This example involves the presentation method mixed fraction incorrectly, it can also be represented in the form general formula:

a bc = a*b+ c / c, where the denominator of the new fraction is formed by multiplying the whole part with the denominator and adding it with the numerator of the original fractional remainder, and the denominator remains the same.

This process also works in reverse side. To separate the whole part and the fractional remainder, you need to divide the numerator of an improper fraction by its denominator using a “corner”.

Multiplying improper fractions produced in a generally accepted way. When writing under a single fraction line, you need to reduce fractions as necessary in order to reduce numbers using this method and make it easier to calculate the result.

There are many helpers on the Internet to solve even complex problems. math problems in various program variations. A sufficient number of such services offer their assistance in counting multiplication of fractions with different numbers in denominators - so-called online calculators for calculating fractions. They are able not only to multiply, but also to perform all other simple arithmetic operations with ordinary fractions And mixed numbers. It’s not difficult to work with; you fill in the appropriate fields on the website page, select the sign of the mathematical operation, and click “calculate.” The program calculates automatically.

Subject arithmetic operations with fractional numbers is relevant throughout the education of middle and high school students. In high school, they no longer consider the simplest species, but integer fractional expressions, but the knowledge of the rules for transformation and calculations obtained earlier is applied in its original form. Well learned basic knowledge give complete confidence in successfully solving the most complex problems.

In conclusion, it makes sense to quote the words of Lev Nikolaevich Tolstoy, who wrote: “Man is a fraction. It is not in the power of man to increase his numerator - his merits - but anyone can reduce his denominator - his opinion about himself, and with this decrease come closer to his perfection.