Level A2 of foreign language proficiency. Abbreviated multiplication formulas. What can you do with an A2 level of English

Abbreviated multiplication formulas.

Studying the formulas for abbreviated multiplication: the square of the sum and the square of the difference of two expressions; difference of squares of two expressions; the cube of the sum and the cube of the difference of two expressions; sums and differences of cubes of two expressions.

Application of abbreviated multiplication formulas when solving examples.

To simplify expressions, factorize polynomials, and reduce polynomials to a standard form, abbreviated multiplication formulas are used. Abbreviated multiplication formulas you need to know by heart.

Let a, b R. Then:

1. The square of the sum of two expressions is the square of the first expression plus twice the product of the first expression and the second plus the square of the second expression.

(a + b) 2 = a 2 + 2ab + b 2

2. The square of the difference of two expressions is the square of the first expression minus twice the product of the first expression and the second plus the square of the second expression.

(a - b) 2 = a 2 - 2ab + b 2

3. Difference of squares two expressions is equal to the product of the difference of these expressions and their sum.

a 2 - b 2 \u003d (a - b) (a + b)

4. sum cube of two expressions is equal to the cube of the first expression plus three times the square of the first expression times the second plus three times the product of the first expression times the square of the second plus the cube of the second expression.

(a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3

5. difference cube of two expressions is equal to the cube of the first expression minus three times the product of the square of the first expression and the second plus three times the product of the first expression and the square of the second minus the cube of the second expression.

(a - b) 3 = a 3 - 3a 2 b + 3ab 2 - b 3

6. Sum of cubes two expressions is equal to the product of the sum of the first and second expressions by the incomplete square of the difference of these expressions.

a 3 + b 3 \u003d (a + b) (a 2 - ab + b 2)

7. Difference of cubes of two expressions is equal to the product of the difference of the first and second expressions by the incomplete square of the sum of these expressions.

a 3 - b 3 \u003d (a - b) (a 2 + ab + b 2)

Application of abbreviated multiplication formulas when solving examples.

Example 1

Calculate

a) Using the formula for the square of the sum of two expressions, we have

(40+1) 2 = 40 2 + 2 40 1 + 1 2 = 1600 + 80 + 1 = 1681

b) Using the formula for the squared difference of two expressions, we obtain

98 2 \u003d (100 - 2) 2 \u003d 100 2 - 2 100 2 + 2 2 \u003d 10000 - 400 + 4 \u003d 9604

Example 2

Calculate

Using the formula for the difference of the squares of two expressions, we obtain

Example 3

Simplify Expression

(x - y) 2 + (x + y) 2

We use the formulas for the square of the sum and the square of the difference of two expressions

(x - y) 2 + (x + y) 2 \u003d x 2 - 2xy + y 2 + x 2 + 2xy + y 2 \u003d 2x 2 + 2y 2

Abbreviated multiplication formulas in one table:

(a + b) 2 = a 2 + 2ab + b 2
(a - b) 2 = a 2 - 2ab + b 2
a 2 - b 2 = (a - b) (a+b)
(a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3
(a - b) 3 = a 3 - 3a 2 b + 3ab 2 - b 3
a 3 + b 3 \u003d (a + b) (a 2 - ab + b 2)
a 3 - b 3 \u003d (a - b) (a 2 + ab + b 2)

(A1-A2-B1-B2-C1-C2)

Cadre européen commun de référence pour les langues: apprendre, enseigner, évaluer (CECRL). Common European competencies of ownership foreign language: learning, teaching, assessment.

Document Council of Europe under the title "Common European Framework of Reference: Learning, Teaching, Assessment" is the result of the work of specialists from the countries of the Council of Europe, as well as representatives of Russia, who systematized the approaches to teaching a foreign language and standardized assessments of language proficiency levels in order to prepare and conduct international examinations in one general rules. “Competencies” allow you to determine what a language learner needs to learn in order to be able to communicate in it, as well as what knowledge and skills are needed to ensure successful communication.

The document defines a standard terminology, a system of units, or a common language to describe what constitutes a subject of study, as well as to describe the levels of language proficiency, regardless of which language is studied and in which educational context (country, educational institution, in courses or privately), and what methods are used.

According to this document, two complexes were introduced: a system of language proficiency levels and a system for describing these levels using standard categories.

The following table represents the levels of ownership:

Do you want to take an English level test right now and find out your level of language proficiency? What are your strengths and what do you still have to learn? We invite you to take an online free test (registration and input is not required) Email), consisting of 60 questions. You will receive the result as soon as you answer the last question.

English level test - instructions

The test determines the level of knowledge of the English language and divides students into 5 groups - from elementary (elementary) to advanced.

The test tests knowledge of language constructions (36 questions) and vocabulary (24 questions). A total of 60 questions must be answered, each of which is given a choice of one of four answers. If you do not know the correct answer to a question and do not mark anything, then the answer on it will be considered incorrect.

There is no time limit for passing the test, but try to keep within 40-45 minutes - this is the time for which this test is designed. For a more accurate assessment of knowledge, it is better not to use dictionaries and textbooks.

Determining the level of English

You can determine the result yourself in accordance with the table below, depending on the number of points you have scored. Read also our articles on how to prepare for and pass international tests: and.

Please note that the English level test only provides estimates and cannot be used for admission to educational institutions. In addition, this test does not assess your writing, reading, or speaking skills.

CEFR level ( Common European Framework of Reference - The level of the Common European Framework of Reference for Languages) is a unified system for assessing language knowledge, using which even knowledge can be compared with each other different languages For example, your English is at B1 level and Chinese is at A2 level.

So let's take the test

TEST (60 questions)

Choose the most appropriate word or phrase for each question

Or in the courses, you will definitely come across the concept of “levels in English” or “English language proficiency levels”, as well as with such obscure designations as A1, B2, and more understandable Beginner, Intermediate and so on. In this article, you will learn what these formulations mean and what levels of language proficiency distinguish, as well as how to determine your level of english.

The levels of English are designed so that language learners can be divided into groups with approximately similar knowledge and skills in reading, writing, speaking and writing, as well as to simplify testing procedures, examinations, for various purposes related to emigration, study abroad and employment. This classification helps in recruiting students into a group and preparing teaching aids, methods, language teaching programs.

Of course, there is no clear boundary between the levels, this division is rather conditional, necessary not so much for students as for teachers. In total, there are 6 levels of language proficiency, there are two types of division:

• Levels A1, A2, B1, B2, C1, C2,
• Beginner, Elementary, Intermediate, Upper Intermediate, Advanced, Proficiency levels.

In fact, these are just two different names for the same thing. These 6 levels are divided into three groups.

Table: English language proficiency levels

The classification was developed in the late eighties - early nineties of the last century, it is fully called the Common European Framework of Reference for Languages: Learning, Teaching, Assessment (abbr. CERF).

English levels: detailed description

Beginner level (A1)

At this level you can:

• Understand and use familiar everyday expressions and simple phrases aimed at solving specific problems.
• Introduce yourself, introduce other people, ask simple personal questions, such as “Where do you live?”, “Where are you from?”, be able to answer such questions.
• Maintain a simple conversation if the other person speaks slowly, clearly, and helps you.

Many who studied English at school speak the language at about the Beginner level. From the vocabulary only elementary mother, father, help me, my name is, London is the capital. You can understand well-known words and expressions by ear if they speak very clearly and without an accent, as in audio lessons for a textbook. You understand texts like the “Exit” sign, and in a conversation with the help of gestures, using individual words, you can express the simplest thoughts.

Elementary level (A2)

At this level you can:

• Understand common expressions common topics, such as: family, shopping, work, etc.
• Talk about simple everyday topics, using simple phrases.
• Tell in simple terms about yourself, describe simple situations.

If at school you had 4 or 5 in English, but after some time did not use English, then most likely you speak the language at the Elementary level. TV shows in English will not be understood, except perhaps for individual words, but the interlocutor, if he speaks clearly, in simple phrases of 2-3 words, in general, you will understand. You can also incoherently and with long pauses for reflection tell the simplest information about yourself, say that the sky is blue and the weather is clear, express a simple wish, make an order at McDonald's.

Beginner - Elementary levels can be called "survival level", Survival English. It is enough to "survive" during a trip to a country where the main language is English.

Intermediate level (B1)

At this level you can:

• Understand the general meaning of distinct speech on general, familiar topics related to everyday life(work, study, etc.)
• Cope with the most typical situations on a trip, travel (at the airport, in a hotel, etc.)
• Write simple connected text on topics that are common or familiar to you personally.
• Retell events, describe hopes, dreams, ambitions, be able to briefly talk about plans and explain your point of view.

Vocabulary and knowledge of grammar is enough to write simple essays about yourself, describe cases from life, write a letter to a friend. But in most cases oral speech lags behind writing, you confuse tenses, think over a phrase, pause to pick up a preposition (to or for?), but you can more or less communicate, especially if there is no shyness or fear of making a mistake.

It is much more difficult to understand the interlocutor, and if it is a native speaker, and even with fast speech and a bizarre accent, then it is almost impossible. However, simple, clear speech is understood well, provided that the words and expressions are familiar. You generally understand if the text is not very complex, and with some difficulty understand the general meaning without subtitles.

Level Upper Intermediate (B2)

At this level you can:

• Understand the general meaning of complex text on concrete and abstract topics, including technical (specialized) topics in your profile.
• Speak quickly enough so that communication with a native speaker occurs without long pauses.
• Write clear, detailed text different topics, explain a point of view, give arguments for and against different points of view on a topic.

Upper Intermediate is already a good, sound, confident command of the language. If you are talking on a well-known topic with a person whose pronunciation you understand well, then the conversation will go quickly, easily, naturally. An outside observer will say that you are fluent in English. However, you can be confused by words and expressions related to topics that are poorly understood by you, all sorts of jokes, sarcasm, allusions, slang.

You are asked to answer 36 questions to test listening, writing, speaking and grammar.

It is noteworthy that to test listening comprehension, phrases like “London is the capital” recorded by the speaker are not used, but short excerpts from films (Puzzle English specializes in learning English from films and TV shows). In English-language films, the speech of the characters is close to how people speak in real life so the test may seem harsh.

Chandler from Friends doesn't have the best pronunciation.

To check the letter, you need to translate several phrases from English into Russian and from Russian into English. The program provides several translation options for each phrase. To test knowledge of grammar, a completely ordinary test is used, where you need to choose one option from several proposed ones.

But you are probably wondering how the program can test the skill of speaking? Of course, an online English proficiency test will not test your speech as a person, but the test developers came up with an original solution. In the task, you need to listen to a phrase from the movie and choose a cue that is suitable for continuing the dialogue.

Talking is not enough, you also need to understand the interlocutor!

The ability to speak English consists of two skills: to understand the speech of the interlocutor by ear and to express one's thoughts. This task, albeit in a simplified form, tests how you cope with both tasks.

At the end of the test you will be shown full list questions with correct answers, you will find out where you made mistakes. And of course, you will see a chart showing your level on a scale from Beginner to Upper Intermediate.

2. Test to determine the level of English with a teacher

To get a professional, “live” (not automated, as in tests) assessment of the level of English, you need English teacher who will test you with assignments and an interview in English.

This consultation is free of charge. First, your city may have Language school, which offers free testing for knowledge of the language and even a trial lesson. Now this is a common practice.

In short, I signed up for a trial test lesson, got in touch on Skype at the appointed time, and the teacher Alexandra and I held a lesson, during which she “tortured” me in every possible way with various tasks. All communication was in English.

My trial lesson on SkyEng. Checking grammar knowledge.

At the end of the lesson, the teacher explained to me in detail in which direction I should develop my English, what problems I have, and a little later she sent a letter with detailed description level of language skills (with marks on a 5-point scale) and methodological recommendations.

This method took some time: three days passed from the application to the lesson, and the lesson itself lasted about 40 minutes. But it is much more interesting than any online test.

Friends! Now I am not engaged in tutoring, but if you need a teacher, I recommend this wonderful site - there are native (and non-native) teachers there 👅 for all occasions and for every pocket 🙂 I myself have gone through more than 50 lessons with teachers that I found there !

The best way to determine if your English language skills are B1 level is to take a quality standardized test. Below is a list of the main internationally recognized tests and their respective B1 scores:

What can you do with knowledge of English at level B1

Level B1 of English will be sufficient to communicate with native English speakers on familiar topics. In the workplace, the B1 level of English allows an employee to read simple reports on familiar topics and write simple emails in their professional area. However, B1 level is not enough to communicate in the workplace only in English.

According to the official CEFR guidelines, a student with B1 English proficiency:

1. Understands the main ideas of clear standard messages on familiar topics regularly encountered at work, school, leisure, etc.
2. Can communicate in most situations that may arise during a stay in a country where the target language is spoken.
3. Can write simple coherent text on topics that are familiar or of personal interest to him.
4. Can describe experiences, events, dreams, hopes and aspirations, state and justify opinions and plans.

More about knowledge of English at level B1

The official conclusions about the student's skills are broken down into smaller sub-items for educational purposes. Such a detailed classification will help you evaluate your own level of English proficiency or help the teacher assess the level of students. For example, a student who speaks English at level B1 will be able to do everything that a student at level A2 can do, as well as:

• to discuss dreams and hopes for the future in the personal and professional sphere. organize and pass an interview when applying for a job in your professional field.
• talk about your TV preferences and favorite programmes.
• describe your education and your plans for future studies.
• talk about your favorite pieces of music and musical trends. Be able to plan an evening of listening to live music.
• talk about management healthy lifestyle life, give and receive advice on healthy habits.
• talk about relationships and acquaintances, including communication with people on social networks.
• visit a restaurant, order food, engage in small talk at dinner, and pay the bill.
• participate in negotiations in their area of ​​expertise, enlisting help in understanding some issues.
• discuss safety issues in the workplace.
• Discuss the norms of polite behavior and adequately respond to impolite behavior.

Of course, progress will depend on the type of course and the individual student, but it can be predicted that the student will achieve a B1 English proficiency level in 400 hours of study (total).