Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ... discussions continue to this day; the scientific community has not yet been able to come to a common opinion on the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a generally accepted solution to the problem..."[Wikipedia, "Zeno's Aporia". Everyone understands that they are being fooled, but no one understands what the deception consists of.
From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.
If we turn our usual logic around, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of “infinity” in this situation, then it would be correct to say “Achilles will catch up with the turtle infinitely quickly.”
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal units. In Zeno's language it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise will crawl a hundred steps in the same direction. During the next time interval equal to the first, Achilles will run another thousand steps, and the tortoise will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein’s statement about the irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.
Another interesting aporia of Zeno tells about a flying arrow:
A flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs to be noted here. From one photograph of a car on the road it is impossible to determine either the fact of its movement or the distance to it. To determine whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to a car, you need two photographs taken from different points in space at one point in time, but from them you cannot determine the fact of movement (of course, you still need additional data for calculations, trigonometry will help you). What I want to draw special attention to is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.
Wednesday, July 4, 2018
The differences between set and multiset are described very well on Wikipedia. Let's see.
As you can see, “there cannot be two identical elements in a set,” but if there are identical elements in a set, such a set is called a “multiset.” Reasonable beings will never understand such absurd logic. This is the level of talking parrots and trained monkeys, who have no intelligence from the word “completely”. Mathematicians act as ordinary trainers, preaching to us their absurd ideas.
Once upon a time, the engineers who built the bridge were in a boat under the bridge while testing the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.
No matter how mathematicians hide behind the phrase “mind me, I’m in the house,” or rather, “mathematics studies abstract concepts,” there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash register, giving out salaries. So a mathematician comes to us for his money. We count out the entire amount to him and lay it out on our table in different piles, into which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his “mathematical set of salary.” Let us explain to the mathematician that he will receive the remaining bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.
First of all, the logic of the deputies will work: “This can be applied to others, but not to me!” Then they will begin to reassure us that bills of the same denomination have different bill numbers, which means they cannot be considered the same elements. Okay, let's count salaries in coins - there are no numbers on the coins. Here the mathematician will begin to frantically remember physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms is unique for each coin...
And now I have the most interesting question: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science is not even close to lying here.
Look here. We select football stadiums with the same field area. The areas of the fields are the same - which means we have a multiset. But if we look at the names of these same stadiums, we get many, because the names are different. As you can see, the same set of elements is both a set and a multiset. Which is correct? And here the mathematician-shaman-sharpist pulls out an ace of trumps from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.
To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I'll show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."
Sunday, March 18, 2018
The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but that’s why they are shamans, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.
Do you need proof? Open Wikipedia and try to find the page "Sum of digits of a number." She doesn't exist. There is no formula in mathematics that can be used to find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics the task sounds like this: “Find the sum of graphic symbols representing any number.” Mathematicians cannot solve this problem, but shamans can do it easily.
Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let us have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.
1. Write down the number on a piece of paper. What have we done? We have converted the number into a graphical number symbol. This is not a mathematical operation.
2. We cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic symbols into numbers. This is not a mathematical operation.
4. Add the resulting numbers. Now this is mathematics.
The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” taught by shamans that mathematicians use. But that is not all.
From a mathematical point of view, it does not matter in which number system we write a number. So, in different number systems the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. With the large number 12345, I don’t want to fool my head, let’s consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We won't look at every step under a microscope; we've already done that. Let's look at the result.
As you can see, in different number systems the sum of the digits of the same number is different. This result has nothing to do with mathematics. It’s the same as if you determined the area of a rectangle in meters and centimeters, you would get completely different results.
Zero looks the same in all number systems and has no sum of digits. This is another argument in favor of the fact that. Question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians nothing exists except numbers? I can allow this for shamans, but not for scientists. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement for numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then this has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical operation does not depend on the size of the number, the unit of measurement used and on who performs this action.
Oh! Isn't this the women's restroom?
- Young woman! This is a laboratory for the study of the indephilic holiness of souls during their ascension to heaven! Halo on top and arrow up. What other toilet?
Female... The halo on top and the arrow down are male.
If such a work of design art flashes before your eyes several times a day,
Then it’s not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort to see minus four degrees in a pooping person (one picture) (a composition of several pictures: a minus sign, the number four, a designation of degrees). And I don’t think this girl is a fool who doesn’t know physics. She just has a strong stereotype of perceiving graphic images. And mathematicians teach us this all the time. Here's an example.
1A is not “minus four degrees” or “one a”. This is "pooping man" or the number "twenty-six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.
Parentheses are used to indicate the order in which actions are performed in numeric, literal, and variable expressions. It is convenient to move from an expression with brackets to an identically equal expression without brackets. This technique is called opening brackets.
Expanding parentheses means removing the parentheses from an expression.
One more point deserves special attention, which concerns the peculiarities of recording decisions when opening brackets. We can write the initial expression with brackets and the result obtained after opening the brackets as an equality. For example, after expanding the parentheses instead of the expression
3−(5−7) we get the expression 3−5+7. We can write both of these expressions as the equality 3−(5−7)=3−5+7.
And one more important point. In mathematics, to shorten notations, it is customary not to write the plus sign if it appears first in an expression or in parentheses. For example, if we add two positive numbers, for example, seven and three, then we write not +7+3, but simply 7+3, despite the fact that seven is also a positive number. Similarly, if you see, for example, the expression (5+x) - know that before the bracket there is a plus, which is not written, and before the five there is a plus +(+5+x).
The rule for opening parentheses during addition
When opening brackets, if there is a plus in front of the brackets, then this plus is omitted along with the brackets.
Example. Open the brackets in the expression 2 + (7 + 3) There is a plus in front of the brackets, which means we do not change the signs in front of the numbers in brackets.
2 + (7 + 3) = 2 + 7 + 3
Rule for opening parentheses when subtracting
If there is a minus before the brackets, then this minus is omitted along with the brackets, but the terms that were in the brackets change their sign to the opposite. The absence of a sign before the first term in parentheses implies a + sign.
Example. Expand the parentheses in the expression 2 − (7 + 3)
There is a minus before the brackets, which means you need to change the signs in front of the numbers in the brackets. In parentheses there is no sign before the number 7, this means that seven is positive, it is considered that there is a + sign in front of it.
2 − (7 + 3) = 2 − (+ 7 + 3)
When opening the brackets, we remove from the example the minus that was in front of the brackets, and the brackets themselves 2 − (+ 7 + 3), and change the signs that were in the brackets to the opposite ones.
2 − (+ 7 + 3) = 2 − 7 − 3
Expanding parentheses when multiplying
If there is a multiplication sign in front of the brackets, then each number inside the brackets is multiplied by the factor in front of the brackets. In this case, multiplying a minus by a minus gives a plus, and multiplying a minus by a plus, like multiplying a plus by a minus, gives a minus.
Thus, the parentheses in the products are expanded in accordance with the distributive property of multiplication.
Example. 2 (9 - 7) = 2 9 - 2 7
When you multiply a bracket by a bracket, each term in the first bracket is multiplied with each term in the second bracket.
(2 + 3) · (4 + 5) = 2 · 4 + 2 · 5 + 3 · 4 + 3 · 5
In fact, there is no need to remember all the rules, it is enough to remember only one, this: c(a−b)=ca−cb. Why? Because if you substitute one instead of c, you get the rule (a−b)=a−b. And if we substitute minus one, we get the rule −(a−b)=−a+b. Well, if you substitute another bracket instead of c, you can get the last rule.
Opening parentheses when dividing
If there is a division sign after the brackets, then each number inside the brackets is divided by the divisor after the brackets, and vice versa.
Example. (9 + 6) : 3=9: 3 + 6: 3
How to expand nested parentheses
If an expression contains nested parentheses, they are expanded in order, starting with the outer or inner ones.
In this case, it is important that when opening one of the brackets, do not touch the remaining brackets, simply rewriting them as is.
Example. 12 - (a + (6 - b) - 3) = 12 - a - (6 - b) + 3 = 12 - a - 6 + b + 3 = 9 - a + b
In this lesson you will learn how to transform an expression containing parentheses into an expression without parentheses. You will learn how to open parentheses preceded by a plus sign and a minus sign. We will remember how to open brackets using the distributive law of multiplication. The considered examples will allow you to connect new and previously studied material into a single whole.
Topic: Solving equations
Lesson: Expanding Parentheses
How to expand parentheses preceded by a “+” sign. Using the associative law of addition.
If you need to add the sum of two numbers to a number, you can first add the first term to this number, and then the second.
To the left of the equal sign is an expression with parentheses, and to the right is an expression without parentheses. This means that when moving from the left side of the equality to the right, the opening of the parentheses occurred.
Let's look at examples.
Example 1. 
By opening the brackets, we changed the order of actions. It has become more convenient to count.
Example 2. 
Example 3.
Note that in all three examples we simply removed the parentheses. Let's formulate a rule:
Comment.
If the first term in brackets is unsigned, then it must be written with a plus sign.
You can follow the example step by step. First, add 445 to 889. This action can be performed mentally, but it is not very easy. Let's open the brackets and see that the changed procedure will significantly simplify the calculations.
If you follow the indicated procedure, you must first subtract 345 from 512, and then add 1345 to the result. By opening the brackets, we will change the procedure and significantly simplify the calculations.
Illustrating example and rule.
Let's look at an example: . You can find the value of an expression by adding 2 and 5, and then taking the resulting number with the opposite sign. We get -7.
On the other hand, the same result can be obtained by adding the opposite numbers of the original ones.
Let's formulate a rule:
Example 1. 
Example 2.
The rule does not change if there are not two, but three or more terms in brackets.
Example 3. 
Comment. The signs are reversed only in front of the terms.
In order to open the brackets, in this case we need to remember the distributive property.
First, multiply the first bracket by 2, and the second by 3.
The first bracket is preceded by a “+” sign, which means that the signs must be left unchanged. The second sign is preceded by a “-” sign, therefore, all signs need to be changed to the opposite
Bibliography
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M.: Mnemosyne, 2012.
- Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. - Gymnasium, 2006.
- Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. - Enlightenment, 1989.
- Rurukin A.N., Tchaikovsky I.V. Assignments for the mathematics course grades 5-6 - ZSh MEPhI, 2011.
- Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. A manual for 6th grade students at the MEPhI correspondence school. - ZSh MEPhI, 2011.
- Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Textbook-interlocutor for 5-6 grades of secondary school. Math teacher's library. - Enlightenment, 1989.
- Online tests in mathematics ().
- You can download those specified in clause 1.2. books().
Homework
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M.: Mnemosyne, 2012. (link see 1.2)
- Homework: No. 1254, No. 1255, No. 1256 (b, d)
- Other tasks: No. 1258(c), No. 1248
That part of the equation is the expression in parentheses. To open parentheses, look at the sign in front of the parentheses. If there is a plus sign, opening the parentheses in the expression will not change anything: just remove the parentheses. If there is a minus sign, when opening the brackets, you must change all the signs that were originally in the brackets to the opposite ones. For example, -(2x-3)=-2x+3.
Multiplying two parentheses.
If the equation contains the product of two brackets, expand the brackets according to the standard rule. Each term in the first bracket is multiplied with each term in the second bracket. The resulting numbers are summed up. In this case, the product of two “pluses” or two “minuses” gives the term a “plus” sign, and if the factors have different signs, it receives a “minus” sign.
Let's consider.
(5x+1)(3x-4)=5x*3x-5x*4+1*3x-1*4=15x^2-20x+3x-4=15x^2-17x-4.
By opening parentheses, sometimes raising an expression to . The formulas for squaring and cubed must be known by heart and remembered.
(a+b)^2=a^2+2ab+b^2
(a-b)^2=a^2-2ab+b^2
(a+b)^3=a^3+3a^2*b+3ab^2+b^3
(a-b)^3=a^3-3a^2*b+3ab^2-b^3
Formulas for constructing an expression greater than three can be done using Pascal's triangle.
Sources:
- parenthesis expansion formula
Mathematical operations enclosed in parentheses can contain variables and expressions of varying degrees of complexity. To multiply such expressions, you will have to look for a solution in general form, opening the brackets and simplifying the result. If the brackets contain operations without variables, only with numerical values, then opening the brackets is not necessary, since if you have a computer, its user has access to very significant computing resources - it’s easier to use them than to simplify the expression.
Instructions
Multiply sequentially each (or minuend with ) contained in one bracket by the contents of all other brackets if you want to get the result in general form. For example, let the original expression be written as follows: (5+x)∗(6-x)∗(x+2). Then sequential multiplication (that is, opening the parentheses) will give the following result: (5+x)∗(6-x)∗(x+2) = (5∗6-5∗x)∗(5∗x+5∗2) + (6∗x-x∗x)∗(x∗x+2∗x) = (5∗6∗5∗x+5∗6∗5∗2) - (5∗x∗5∗x+5∗ x∗5∗2) + (6∗x∗x∗x+6∗x∗2∗x) - (x∗x∗x∗x+x∗x∗2∗x) = 5∗6∗5∗x + 5∗6∗5∗2 - 5∗x∗5∗x - 5∗x∗5∗2 + 6∗x∗x∗x + 6∗x∗2∗x - x∗x∗x∗x - x ∗x∗2∗x = 150∗x + 300 - 25∗x² - 50∗x + 6∗x³ + 12∗x² - x∗x³ - 2∗x³.
Simplify the result by shortening the expressions. For example, the expression obtained in the previous step can be simplified as follows: 150∗x + 300 - 25∗x² - 50∗x + 6∗x³ + 12∗x² - x∗x³ - 2∗x³ = 100∗x + 300 - 13∗ x² - 8∗x³ - x∗x³.
Use a calculator if you need to multiply containing only numerical values, without unknown variables. Built-in software
I am continuing the series of methodological articles on the topic of teaching. It's time to consider the features of individual work math tutor for 7th grade students. It is with great pleasure that I will share my thoughts on the forms of presentation of one of the most important topics in the 7th grade algebra course - “opening parentheses.” In order not to try to grasp the immensity, let’s stop at its initial stage and analyze the tutor’s method of working with multiplying a polynomial by a polynomial. How math tutor acts in difficult situations when weak student does not accept the classical form of explanation? What tasks should be prepared for a strong seventh grader? Let's consider these and other questions.
It would seem, what’s so complicated about this? “Brackets are as easy as shelling pears,” any excellent student will say. “There is a distribution law and properties of powers for working with monomials, a general algorithm for any number of terms. Multiply each by each and bring similar ones.” However, not everything is so simple when working with laggards. Despite the efforts of the math tutor, students manage to make errors of all sizes even in the simplest transformations. The nature of the errors is striking in its diversity: from small omissions of letters and signs to serious dead-end “stop errors”.
What prevents a student from completing the transformations correctly? Why is misunderstanding possible?
There are a huge number of individual problems, and one of the main obstacles to the assimilation and consolidation of material is the difficulty in timely and quick switching of attention, the difficulty in processing a large amount of information. It may seem strange to some that I am talking about a large volume, but a weak 7th grade student may not have enough memory and attention resources even for four terms. Coefficients, variables, degrees (indicators) interfere. The student confuses the order of operations, forgets which monomials have already been multiplied and which remained untouched, cannot remember how they are multiplied, etc.
Numerical Approach for Math Tutor
Of course, you need to start with an explanation of the logic behind the construction of the algorithm itself. How to do it? We need to pose a problem: how to change the order of actions in an expression 
so that the result does not change? I quite often give examples that explain how certain rules work using specific numbers. And only then I replace them with letters. The technique for using the numerical approach will be described below.
Motivation problems.
At the beginning of a lesson, it is difficult for a math tutor to gather a student if he does not understand the relevance of what is being studied. Within the syllabus for grades 6–7, it is difficult to find examples of using the rule for multiplying polynomials. I would emphasize the need to learn change the order of actions in expressions The student should know that this helps solve problems from experience in adding similar terms. He had to add them together when solving equations. For example, in 2x+5x+13=34 he uses that 2x+5x=7x. A math tutor simply needs to focus the student’s attention on this.
Math teachers often refer to the technique of opening parentheses as "fountain" rule.
This image is well remembered and should definitely be used. But how is this rule proven? Let us recall the classical form, which uses obvious identity transformations:
(a+b)(c+d)=(a+b) c+(a+b) d=ac+bc+ad+bd
It is difficult for a math tutor to comment on anything here. The letters speak for themselves. And a strong 7th grade student doesn’t need detailed explanations. However, what to do with the weak, who point-blank does not see any content in this “literal jumble”?
The main problem that interferes with the perception of the classical mathematical justification of the “fountain” is the unusual form of writing the first factor. Neither in the 5th grade nor in the 6th grade did the student have to drag the first bracket to each term of the second. Children dealt only with numbers (coefficients), most often located to the left of the brackets, for example:
By the end of 6th grade, the student has formed a visual image of an object - a certain combination of signs (actions) associated with brackets. And any deviation from the usual view towards something new can disorient a seventh grader. It is the visual image of the “number + bracket” pair that the math tutor uses when explaining.
The following explanation can be offered. The tutor reasons: “If there was some number in front of the bracket, for example 5, then we could change the procedure in this expression? Certainly. Then let's do it 
. Think about whether his result will change if instead of the number 5 we enter the sum 2+3 enclosed in brackets? Any student will tell the tutor: “What difference does it make how you write: 5 or 2+3.” Wonderful. You will get a recording. The math tutor takes a short break so that the student visually remembers the picture-image of the object. Then he draws his attention to the fact that the bracket, like the number, “distributed” or “jumped” to each term. What does this mean? This means that this operation can be performed not only with a number, but also with a parenthesis. We got two pairs of factors and . Most students easily cope with them on their own and write the result to the tutor. It is important to compare the resulting pairs with the contents of the brackets 2+3 and 6+4 and it will become clear how they open.
If necessary, after the example with numbers, the math tutor conducts a letter proof. It turns out to be a cakewalk through the same parts of the previous algorithm.
Formation of the skill of opening brackets
Forming the skill of multiplying parentheses is one of the most important stages of a math tutor’s work with a topic. And even more important than the stage of explaining the logic of the “fountain” rule. Why? The rationale for the changes will be forgotten the very next day, but the skill, if it is formed and consolidated in time, will remain. Students perform the operation mechanically, as if retrieving a multiplication table from memory. This is what needs to be achieved. Why? If every time a student opens parentheses he remembers why it is opened this way and not otherwise, he will forget about the problem he is solving. That is why the math tutor devotes the remaining time of the lesson to transforming understanding into rote memorization. This strategy is often used in other topics.
How can a tutor develop the skill of opening parentheses in a student? To do this, a 7th grade student must complete a number of exercises in sufficient quantities to consolidate. This raises another problem. A weak seventh grader cannot cope with the increased number of transformations. Even small ones. And mistakes fall one after another. What should a math tutor do? Firstly, it is recommended to draw arrows from each term to each one. If a student is very weak and is not able to quickly switch from one type of work to another, or loses concentration when following simple commands from the teacher, then the math tutor himself draws these arrows. And not all at once. First, the tutor connects the first term in the left parenthesis with each term in the right parenthesis and asks them to perform the corresponding multiplication. Only after this the arrows are directed from the second term to the same right bracket. In other words, the tutor divides the process into two stages. It is better to maintain a short time pause (5-7 seconds) between the first and second operations.
1) One set of arrows should be drawn above the expressions, and the other below them.
2) It is important to skip between lines at least a couple of cells. Otherwise, the recording will be very dense, and the arrows will not only climb onto the previous line, but will also mix with the arrows from the next exercise.
3) In the case of multiplying brackets in the format 3 by 2, arrows are drawn from the short bracket to the long one. Otherwise, there will be not two, but three of these “fountains”. The implementation of the third is noticeably more complicated due to the lack of free space for the arrows.
4) arrows always point from the same point. One of my students kept trying to put them side by side and this is what he came up with:
This arrangement does not allow selecting and recording the current term with which the student works at each stage.
Tutor's finger work
4) To keep attention on a separate pair of multiplied terms, the math tutor puts two fingers on them. This must be done in such a way as not to block the student’s view. For the most inattentive students, you can use the “pulsation” method. The math tutor moves his first finger to the beginning of the arrow (to one of the terms) and fixes it, and with the second he “knocks” at its end (to the second term). Ripple helps to focus attention on the term by which the student is multiplying. After the first multiplication by the right parenthesis is completed, the math tutor says: “Now we work with the other term.” The tutor moves the “fixed finger” towards it, and runs the “pulsating” finger over the terms from the other bracket. The pulsation works like a “turn signal” in a car and allows you to focus the attention of an absent-minded student on the operation he is performing. If the child writes small, then two pencils are used instead of fingers.
Repetition optimization
As when studying any other topic in an algebra course, multiplying polynomials can and should be integrated with previously covered material. To do this, the math tutor uses special bridge tasks that allow you to find the application of what you are studying in various mathematical objects. They not only connect topics into a single whole, but also very effectively organize the repetition of the entire mathematics course. And the more bridges the tutor builds, the better.
Traditionally, 7th grade algebra textbooks integrate opening parentheses with solving linear equations. At the end of the list of numbers there are always tasks of the following order: solve the equation. When opening the brackets, the squares are reduced and the equation is easily solved using 7th grade tools. However, for some reason, the authors of textbooks conveniently forget about constructing a graph of a linear function. In order to correct this shortcoming, I would advise mathematics tutors to include parentheses in analytical expressions of linear functions, for example. In such exercises, the student not only trains the skills of carrying out identical transformations, but also repeats graphs. You can ask to find the point of intersection of two “monsters”, determine the relative position of the lines, find the points of their intersection with the axes, etc.
Kolpakov A.N. Mathematics tutor in Strogino. Moscow