What will become of the square of a rectangular arkush paper. Zastosuvannya elementіv trez pіd hr lessons of mathematics. Squares from Chotiriokh chastyn

Distributed: Maths

meta lesson:

• Uzagalnennya that systematization of taking away knowledge.
• Expansion of the statement of studies about the rozvyazannya task of rebuking the largest and smallest values.

Hid lesson

1 Stage lesson

Introductory teacher: skin people sometimes appear in the situation, if you need to know best way rozv'yazannya be-what tasks.

For example: technologists try to organize the design in such a way, so that they can choose the best products, designers want to plan the fittings on the spacecraft in such a way, so that the fittings will be the smallest, etc.

It can be boldly said that the task of knowing the highest and lowest values ​​can be practical zastosuvannya.

To prove my words, I would like to quote L.N. Tolstoy "What a lot of people need the land" about the peasant Pakhom, who bought the land from the Bashkirs.

- What will be the price? - like Pahom.
- We have one price: 1000 UAH. per day.
I don’t understand Pahom.
- What kind of tse zahid is a day? How many tithes will he have?
- My tsyogo, - it seems, - I can’t rahuvati. And we sell in a day; If you buy skils for a day, then it's yours, and the price is 1000 UAH.
Zdivuvavsya Pahom.
- That tse, - it seems, - the day you will get around the earth will be rich.
The foreman laughed.
- Mustache is yours, - it seems. - Only one thought: if you don’t come back the day before that month, for whatever you take, your pennies arose.
- And how, - it seems Pahom, - means, where will I go?
- And we will stand on the spot, children will love you; we stand at the same time, but go, rob the colo, and take the scraper with you, de need, mark, on the pits of the hole, put the turf; Let's go from hole to hole with a plow. Whatever you want, take a stake, just before sunset, come before that month, from which you took it. What will you do, everything is yours.

The figure that Pakhom had seen is a little one. What is that figure? (rectangular trapezium)

Power supply: How do you care, what is the largest area in Pahom. (because of the urance of the fact that the dilyanki sound to make the shape of a chotirikutnik)? Today at the lesson mi tse z'yasuёmo.

In order to accomplish this task, do we need to guess what steps we need to take in order to accomplish extreme tasks?

1. The task is transferred to my function.
2. By means of analysis, the most and least values ​​are found.
3. Z'yasuvati, what a practical sense can take away the result.

Manager No. 1 (Virishimo all class)

The perimeter of the rectangle is 120 cm.

Turning to the task, for which the lesson was learned. What is the largest area of ​​\u200b\u200bcovering Pakh (for the sake of the fact that the villagers sound like a chotirikutnik)? It is discussed with the teachings that I will take the largest area of ​​the moment to cover Pahom.

2 Stage lesson

For the end of the day on the doshtsi we will write the instructions for an explanation (їх two).

Manager No. 1

To know, for some minds, the cost of tins in the preparation of cans of a cylindrical shape of a given capacity will be the smallest.
I have great respect for the lads, that hundreds of millions of cans are being produced in our country, and woolen wool has been protected, I would like to allow additional production of millions of cans by 1%.

Manager No. 2

Chovni roztashovani on vіdstanі 3 km from the nearest point A of the coast. At point B, which is located on the vіdstanі 5 km. look A, wait. Chovnyar asks to come to the rescue, for that you need to eat there at the shortest hour. Chauvin collapses from the speed of 4 km/year, and the passenger 5 km/year. To what point on the shore can a chovnyar moor?

3 Stage lesson

Robot z group іz advancing zakhist zavdan.

Manager No. 1

One of the faces of a rectangular parallelepiped is a square. The sum of the lengths of the edges that emerge from one of the vertices of the parallelepiped is 12. Find the largest possible volume.

Manager No. 2

For installation, a support with a volume of 240 dm3 is required for the shape of a rectangular parallelepiped. The base of the stand, as it will be mounted in the base, is a straight cut. The length of the rectangle is larger than the width. The rear wall of the stand will be mounted at the wall of the workshop. When mounting the stand, the walls, not mounted in the basement or the wall, join together for additional support. Consider expanding the supports, if the crown of the welded seam will be the smallest.

Manager No. 3

From a round deck, a beam with a straight-cut beam of the largest area is cut. Find out the size of the cross section of the beam, so the radius of the cross section of the deck is 30 cm.

Manager No. 4

From a rectangular sheet of cardboard with sides of 80 cm and 50 cm, it is necessary to make a box of a rectangular shape, bending along the edges of the square and bending the edges, which are settled. What height can the box be, so the volume is the largest. Know tsey obsyag.

4 Stage lesson

Appointment for the evaluation of the selection.

Manager No. 1

For a dart with a length of 80 cm, you need to build a rectangle of the largest area. Know yoga razmіri.

Manager No. 2

The sum of the dozhins of the ribs of a regular triangular prism is 18√3. To know the greatest possibility of such a prism.

Manager No. 3

The diagonal of a rectangular parallelepiped, one of the side faces of which is a square, is older than 23. Find the greatest possible length of such a parallelepiped.

5 stage lesson

Side 6 of 8

Brozdіl p'yaty.

DISCLAIMER OF FIGURES. ROZDIL I

In this offensive division, we can easily develop a wealth of miraculous geometric paradoxes. All the smells start from the cutting of the figures into shmatki and end with the folding of these shmatkiv in a new figure. When this happens, the damage is formed, that part of the primary figure (it can be a part of the area of ​​the figure, or one of the many images on the small ones) of absence has arisen. If the shmatki turn on their cobs, a part of the square, or the little ones, that have arisen, in a taєmnichiy rank, they blame again.

The geometrical nature of these tsіkavih zniknen and appearing in truth zarahuvannya tsikh paradoxіv up to the range of mathematical puzzles.

The paradox of lines

All the numerical paradoxes that we choose to look at here are based on the same principle, which we call the “principle of attached re-experimentation”. The axis is already an old and elementary phenomenon, which clearly explains the essence of this principle.

Ten vertical lines of the same length are laid on a rectangular arch paper and drawn by a diagonal dotted line, as shown in fig. fifty.

We look at the vіdrіzki tsikh lines above the diagonal and below it; It doesn’t matter to remember that the first one is changing, and the other is obviously getting bigger.

We cut the rectangle along the dotted line and break the lower part to the left down, as shown in fig. 51.

Porahuvavshi a number of vertical lines, you will see that now there are nine. How did the line come and where? Push the left part in the forward position, and the line will appear again.

But how did the line stand in its place and the stars came from?

The heart of the food is made mysterious, but after a little thought, we become aware that the line is no longer okrema when we don’t know and don’t blame. It seems like this: the increase in the growth is exactly the same as the length of the dermal line from the cob.

Possibly, the essence of the paradox appears more differently, like yogo іlustruvati on the fireplaces.

Take five kopecks of stones for chotiri stones in the merchant. Let's move one fireplace from another to the first, two fireplaces from the third to the other, three from the fourth to the third and, nareshti, all four fireplaces from the fifth to the fourth. Rice. 52 explain our actions.

After such a change, it appears that there are no more cups than chotiri. It’s impossible to ask for food, as the cup has come, to the one that the stones were redistributed in such a way that in the skin of some cups the stone was finished. Exactly the same is seen in the phenomena with the lines. If the parts of the sheet are folded diagonally, the cuts of the cut lines are redistributed and the skin line, which comes out at the same time, becomes a troch for the cob.

Signature of an individual

Let's move on to a description of the ways in which paradoxes with lines can be worked out tsіkavіshim and tsіkavim. What can, for example, be achieved by replacing the appearance of lines with such appearances and the appearance of flat figures. Here, especially, images of olives, cigarettes, caegles, capelyukhs with a high thulle, flasks with water and other vertically extended objects, the nature of the image of which until and after the sound of zsuvu are left the same. For some artistic guilt, you can take folding objects. To marvel, for example, at the accusations that you know, in fig. 53.
With the sound of the lower smuga on the upper part of the little one, to the left, all the droplets become uncoupled, only one person appears! (Div. lower part of the little one). Silently feed, as if it were the same guise, the shards, when the chotir is destroyed, the individuals are divided into two parts. The same part of Potim is uprooted, I will focus on the leatherette of the Oddatatkovikh rice: one, set, bilsh dovgiy nis, іnshe - knitting pidboriddy, i.e., one of the little one, and the tsygos, zealo, zealino, zyl shmatka line.

"Znickly Warrior"

In this puzzle, the paradox with lines is given a circular shape and rectilinear ribs are replaced by figures of 13 warriors (Fig. 54).
The great arrow points out when you want to go to the pivnіchniy skhіd. As soon as the little ones are split on the stake, and then we turn the inner part against the year’s arrow, then the figures are divided into parts, then we come back again, but in a different way, and if the arrow is large in the same way, the little one will be S.Z. 12 warriors (Fig. 55).
When wrapping a stake at a straight line to the position, if the big arrow rises again in the NE, the known warrior will appear again.

Yakscho fig. 54 look more respectfully, you can remember that two warriors in the left lower part of the little one are ruffled in a special way: the stench is one against one, just like all the others are placed in a lancet. The two figures correspond to the extreme lines of the phenomena with the vertices. Vyhodyachi s could be a little one, in the skin of these figures, but part of the leg is to blame, and in the turned position of the wheel, this little bit is less commemorated, it was better to depict their instructions.

Significantly more, that the image of a little girl with a significantly greater guilt, lower can be seen from the first glance. So, for example, if the figures were left in a vertical position in all parts of the globe, it is necessary in one way for the mother to protect the left foot of the right, and in another, on the other hand, to protect the right left foot.

wicked rabbit

The paradox of vertical lines can obviously be shown in more foldable objects, for example in human forms, the figures of creatures are thin. bud. On fig. 56 shows one option.
If, after splitting along the comrade line, the rectilinear A, one rabbit knows, depriving himself of an Easter egg. As a replacement for the permutation of the rectangles A and B, to distinguish the right half of the baby along the dotted line and remember the right parts with the help of the right parts, the number of rabbits increases to 12, however, one rabbit eats the ear and shows other funny details.

Razdіl shosty.

DISCLAIMER OF FIGURES. ROZDIL I I

The paradox of the cheater

At a close link with paradoxes, looked at in front of the division, it goes through another class of paradoxes, at which the “principle of attached re-exploitation” explains the secret appearance of the appearance of the area. One of the oldest and simplest examples of paradoxes of this kind is shown in fig. 57.
Shakhova doshka razrisaetsya navskis, as shown on the left half of the little one, and then part of it sways to the left down, as shown on the right half of the little one. Like a tricot, which protrudes at the upper right kutka, cut it with scissors and place it in the free space, which may look like a tricutnik in the lower left kut, a little one, then weide a straight cut in 7x9 square units.

Pochatkova area was 64 square units, now there are 63 square units.

It seems clear that our diagonal line should go lower than the left lower kut of the cell, as it is located at the upper right kut of the doshka.

Zavdyaki tsomu vіdrіzanіy trikutnik maє vysotu, scho dorіvnyuє not 1, but 1 1/7. I, in this rank, the height is not equal to 9, but 9 1/7 to one. The increase in height on 1/7 of the single unit is unremarkable, but, taking it to the point of respect, it is not necessary to bring it up to the required area of ​​the rectangle in 64 square units.

The paradox becomes even more clear, for example, instead of the checker's board, take just a square arch of paper without a cleat, because in our case, with a respectful veil, it appears inaccurately zmikannya clitin uzdovzh line rose.

The link between our paradox and the paradox of vertical lines, we will look at the front split, we will understand, as if walking behind the curtains of the line. When protruding the line of the rose, the upland appears, that over the line the parts of the rose clitins (for a little stench are darkened) step by step change, and along the line step by step increase. There were fifteen darkened cleats on the shahivnitsa, and only fourteen on the straight-cut, after rearranging the parts, there were only fourteen. The appearance of one darkened clitiny, which is given, is simply another form of the paradox that has been looked at. If we vіdrіzaєmo and potіmіshaєmo small trikutnik, we actually cut the part of A shakhіvnitsі into two shmatki, yakі then minyayutsya mіstsy vzdovzh diagonally.

For the puzzle, only the clitiny is important, which fit to the line of the cut, the solution cannot have any meaning, playing the role of decoration. However, their presence changes the nature of the phenomenon. Deputy of one of the few small clitins (otherwise there are folded figures, say, gradient maps, human beings, etc., which can be baptized in the middle of the skin cells) sticking here behind the serpentine area of ​​the great geometric figure.

Paradox from the Square

The axis is one more paradox from the plane. Changing positions A and Z, as shown in fig. 58, it is possible to convert a rectangle with an area of ​​30 square ones into two smaller rectangles from a zagal area of ​​32 square ones, taking, in this order, “winning” from two square ones. Like in the frontal paradox, here the role is played only by clitiny, which adjoin to the line of rose. Other needs are less like formalized.
For this phenomenon, two different methods are used to draw figures into pieces.

You can start from a great rectangle with a size of 3x10 singles (the upper part of Fig. 58), carefully drawing in a new diagonal, then two smaller rectangles (the lower part of Fig. 58) will be 1/5 singles shorter than their dimensions, which are created.

Ale can be almost made of figures, folded from two neatly leaning smaller rectangles in the size of 2x6 and 4x5 single; The same twists that connect point X with point і point Y with point Z do not become a straight line. Only to the fact that they make a blunt kut with a peak at the point U, which is close to a burnt one, laman XUZ is created in a straight line. Therefore, the figure, folded from parts of small rectangles, will not be a rectangle, so that the parts will be slightly twisted diagonally. A paradox with a shahіvnice, so the very same as and more other paradoxes, as we choose to look at in this division, can also be presented in two variants. In one of them, the paradox is to go beyond the rahunka of an insignificant change, or to increase the height (or width) of the figures; we're not bothered to chatter.

Changing the shapes and sharpening the diagonals, this paradox can be given a different design. You can reach it either by increasing the area by 1 square unit or by 2, 3, 4, 5 units, etc.

Option from a square

In one thinned variant of the windows, the rectangles with a size of 3x8 and 5x8 alone, being assigned one to one, make a great checkerboard in 8x8 cells. Qi rectangles are cut into parts, as if they were re-arranged to establish a new great rectangle with an increase in area, which is given, in one square unit (Fig. 59).
The essence of the phenomenon lies in the attack. With a neat posture of the armchair of the square of the Suvoro diagonal of the great rectangle, do not go out. The deputy of her is a rhomboid-like figure, the floorings are twisted so that the sides are made such that they were angry. On the other side, with a neatly drawn diagonal of the great rectangle; the height of the upper one of the two rectangles, which will become a square, will be three times larger, the lower one may be buti, and the lower rectangle - three times wider. It is worth noting that the inaccurate zmikannya of the parts of the figure with a different method of drawing falls more in the first one; the first way is the shortest. Like in the butts, which were previously scribbled, in the middle of the clitin, rossichennyh diagonally, you can paint mugs, physiognomy, or maybe some figures; when rearranging the storage parts of the rectangles in these figures, we make one more and one less.

Fibonacci numbers

It appears that the two sides of the two parts, which become figures (Fig. 59 and 60), are members of the Fibonacci series, that is a series of numbers that starts from two ones: 1, 1, skin from those, starting from the third, and the sum of the two in front . Our row looks like 1, 1, 2, 3, 5, 8, 13, 21, 34…
The roztashuvannya of the parts, on a square, looking like a rectangle, illustrating one of the powers of the Fibonacci series, and stepping on it: when squared, be it a member of that row, there will come out two additional members of the row, plus or minus one. In our butt, the side of the square is 8, and the area is 64. The number in the Fibonacci row is stashed between 5 and 13. The shards of the number 5 and 13 become the dozhins of the sides of the rectangle, then the square is due to equal 65, which gives one increase.

The stars of this power in a row can be called a square, the side of which is the number of Fibonacci, more than one, and then we can cut it up to two forward numbers of the row.

For example, take a square 13x13 singles, then divide the three sides of the next into sections with a length of 5 and 8 singles, and then split it, as shown in fig. 60. The area of ​​the square is 169 square units. The sides of the rectangle, covered with parts of squares, will be 21 and 8, which gives an area of ​​168 square units. Here, the corners of the parts of the bridles of the diagonal are twisted, one square unit is not added, but is used.

If you take a square with side 5, then you will end up spending one square unit. Can you formulate blatant rule: Having taken for the b_k of the square the number of the “first” subsequence of the expansions of the Fibonacci numbers (3, 8 ...) in one, and having added the rectangle to the parts of the square, we take away the number of the “first” diagonal of the enlightenment and like the succession of the increase in the area, on one alone. Taking the number z of the “other” subsequence (2, 5, 13 ...) as the square of the square, we take away the diagonal of the rectangle overlapping the area and the cost of one square unit of the area.

You can inspire the phenomenon on the square with the side in two alone. But then in a 3x1 rectangle, the floorings are obviously overlapped, so the effect of the phenomenon is completely absorbed.

Vykoristovuyuchi for the phenomenon of the іnshі series of Fibonacci, you can take impersonal options. So, for example, squares, based on rows 2, 4, 6, 10, 16, 26, etc., lead to an increase in the area of ​​4 square units. The value of these costs or growths can be determined by calculating for this row the difference between the square, whether it be a yogo member and the work of two yogo suicidal members, is evil and right-handed. Row 3, 4, 7, 11, 18, 29, etc. gives an increase or an expense of five square ones. T. de Mulidar nav_v small squares, based on rows 1, 4, 5, 9, 14, etc. The side of this square is taken equal to 9, and after the transformation of yoga into a rectangle, 11 square units are used. Row 2, 5, 7, 12, 19 ... also gives a waste of 11 square units. In both directions, the overlap (or enlightenment) of the diagonals shows larger floorings, which can be remembered at once.

Having denoted the three last Fibonacci numbers through A, B and C, and through X - the cost or the increase in area, we take two formulas:

A + B = C

Y 2 \u003d AC ± X

If you give a substitute for X money increase, or a cost, and replace the number, as it is taken for the length of the side of the square, then you can induce square alignment, for which two other Fibonacci numbers are known, although, obviously, they will not be obligatory rational numbers. It appears, for example, that, adding a square to figures with rational lengths of the sides, it is impossible to take an increase or an expense in two or three square units. For help irrational numbers tse, zvіsno, you can reach. Thus, the Fibonacci series 2 1/2 , 2 2 1/2 , 3 2 1/2 , 5 2 1/2 gives an increase or loss in two square units, and the series 3 1/2 , 2 3 1/ 2, 3 3 1/2, 5 3 1/2 increase to increase or spend in three square units.

Option from a straight cut

There are a lot of ways in which a rectangle can be cut into a small number of parts, and then folded into a larger rectangle looking like a smaller area. On fig. 61 shows the phenomenon, as well as the foundations on the Fibonacci series.
Similar to a well-glanced ridge with a square, the choice of some kind of Fibonacci number with “other” subsequence, like the width of the first rectangle (in this ridge 13) is reduced to an increase in the area of ​​another rectangle by one square unit.

If we take the Fibonacci number from the “additional” subsequence for the width of the first rectangle, then the area of ​​another rectangle will change by one unit. The loss of that increase in area is explained by small overlaps with gaps of the diagonal cut of another rectangle. The second version of such a rectangle of indications in fig. 62 when prompted by another rectangle, build up to an increase in area on two square units.

As if the shaded part of the area of ​​another rectangle is placed over the unshaded part, two diagonal cuts are merged into one great diagonal. Now rearranging the parts A and B (like in Fig. 61), we take another rectangle with a larger area.

Another version of the paradox

When subsuming the area of ​​the parts, the permutation of tricots and C at the upper part of Fig. 63 to produce until you spend one square unit, which is given.
As a reader to respect, you look at the square of the area of ​​​​shaded parts: on the upper part of the little one there are 15 shaded squares, on the bottom - 16. Now we have a rectangle in front of us, which can be divided into 5 parts, and then, changing them with parts, we will fold a new rectangle, moreover, without regard for those that have too many linear cuts, the middle ones are open one square per unit) (64) .
Possibility of transforming one figure into a foreign country, quiet of the very same roses, ale with an opening in the middle of the perimeter, running on the step. If you take the point X exactly in three units from the base and in five units from the side of the rectangle, then the diagonal will not pass through it. However, the laman, which connects point X with the opposite vertices of the rectangle, moves so little in the direction of the diagonal that it will be unremarkable.

After permutation of tricots B and C on the lower half, a small part of the figure is slightly twisted diagonally.

On the other side, as if at the upper part of the little one, you can see the line, which is behind the proliferating peaks of the rectangle, as if the diagonal was drawn exactly, the XW line will be three times longer than three alone. And as a result of this, another rectangle will be cheap, lower it is. At the first slope, a single area, which is defective, can be divided from kuta to kut and make it possible to cross the diagonals. The other side has a square of divisions along the width of the rectangle. As we already know from the foregoing, all similar paradoxes can be seen in up to one of two options. In both cases, the inaccuracies of the figures of the flooring are insignificant, but the stench is unremarkable.

The most thinned form of this paradox is the squares, as if they were redistributed, the parts of that adorned lapel are filled with squares.

Such squares are available in unidentified variants and with openings, whether there are a lot of square ones. Deyakі, naytsіkavіshi їх images in fig. 65 and 66.

You can show a simple formula that connects the size of the opening with the proportions of the great tricutnik. Three razmіri, about yaki timetsya, mi meaningfully through A, B to C (Fig. 67).
I will open the area in square units to the difference between the creation A to Z and the nearest multiple to the new multiple of B. So, in the remaining butt, the addition of A and C is 25. The closest multiple of B to 25 є 24, so it’s open to go into one square unit. This rule is not independent, since the right diagonal is drawn, or the point X in fig. 67 is neatly applied on the linen of a linear square grid.

As if the diagonal, as if it were possible, swayed like a perfectly straight line, or as the point X was taken exactly from the vertices of the square grid, then no phenomenon would appear. In these cases, the formula gives an opening of zero square units, signifying zim, sig- nificantly , that open dumbly.

Option from tricoutnik

Let's turn to the first butt of the phenomenon (div. Fig. 64). It is respectful that the great tricouter A does not change his camp, even as other parts move. Shards of this tricoutnik do not play a significant role in the paradox, it can be thrown off, overfilling the right tricoutnik, cutting into chotiri parts. The qi parts can then be redistributed, otrimuyuchi at the same time a straight-cut tricot with a lapel (Fig. 68), nіbi dovnyuє vyhіdny.
Folding two such straight-cut tricots with legs, you can induce rich options for equal-femoral tricots, similar to the one shown in fig. 69.
So, just like in earlier paradoxes, the tricots can be done in two ways: either draw the side lines strictly straight, then point X cannot be put on the line of the square grid, or place point X exactly on the line, then the side sides will be slightly swollen or ugly. The rest method, it turns out, better masks the inaccuracies of the armchair. The paradox will be even more wondrous, like on the parts, to make a tricot, to apply lines of a square grid, supporting it, so that the parts were prepared with the necessary accuracy.

Giving our equal-femoral tricots a wide range, you can achieve an increase in spending any pair of square ones.

A deck of typical applications is given in fig. 70, 71 and 72.

Putting together the foundations of two equal-femoral tricots, whether of any of these types, you can indulge yourself different options diamond mind; however, the stench does not add anything truly new to our phenomenon.

Squares from Chotiriokh chastyn

Mustaches have been looked at by us to see the paradoxes from the changing area close to each other for the way of awakening. Prote іsnuyut paradoxes, otrimanі th duzhe vіdminnimi methods. It is possible, for example, to cut a square into several parts of the same shape to the world (Fig. 73), and then fold them in a new way, as shown in Fig. 74. At whom the square comes out, the rozmіri of whom are made such that they did not change and at the same hour with an opening in the middle.
In a similar rank, you can cut a rectangle with some kind of spivvіdnoshennya dovzhin storіn. Tsikavo, like point A, in which two are tinkering, they are such that they did not change at the same time with an opening in the middle.

In a similar rank, you can cut a rectangle with some kind of spivvіdnoshennya dovzhin storіn. Tsikavo, scho point A, in which two mutually perpendicular lines of the cut are intertwined, may be in some place in the middle of the rectangle. In the skin case, when the parts are redistributed, they show an opening, and the rosemary is deposited according to the size of the kuta, made by the lines of the cut from the sides of the rectangle.

This paradox is reminiscent of its relative simplicity, the prote vin is richly savored by the fact that it can be seen with a superficial twist that the sides of the other rectangle may be three times larger, the lower sides of the first one.

Larger folding method of cutting a square into pieces, with which the inner opening comes out, images in fig. 75.

Vіn foundations on the paradox of shahіvnice, yakim vіdkrivаєtsya pravzhnіy rasdіl. Respectfully, it is necessary to turn over the back side of the fire to redistribute the parts of the two. It is also worth respecting that with the removal of parts A, we need a straight-cut tricout, folding from three parts, the middle of which can be made open.

Squares from three parts

What is the best way to cut a square into three parts, how can it be folded in a new way, so that a square with an opening in the middle? Vidpovid be positive. One thin solution is rooted in a fixed paradox, looked at in front of the division.

Instead, in a special way, arrange the pictures in ledges, and cut them in a straight line (horizontally), place the pictures on one straight line, and cut them in ledges. The result is striking: not only does the picture disappear, but on the spot it appears to open.

Square from two parts

Chi can you rob those yourself with two parts?

I don’t think that in any case it is possible in some way to take the inner opening off the square for the chest of an incomparable increase in height and width. However, it has been shown that the paradox with an opening at the square, which is divided into two parts, can be inspired by the principle, which is stagnant in the paradox of the warrior, which is known. In this way, the place of placement of figurines in a spiral, or in a gathering of them, is placed on a stake, then it is split in a spiral or steps; in the rest of the day, the vin may look like a gear wheel from the teeth of different sizes. When wrapping the wheel, one figurine appears and its replacement is revealed.

Unruly and wrapping parts are carefully fitted one to one only in position, if they are open. In the external position, small lumen is visible near the skin tooth, which is roaring of the feet, or one without interruption all-round lumen when it is roamed, which goes in a spiral.

Even though the outer rectangle is not a square, it can be cut into two parts, and then we will cut it in the middle of the opening with a little memory of the outer cuts. On fig. 76 shows one option.

Offenses are part of each other, both for the form, and for the dimensions. The simplest way to demonstrate this paradox is to follow the next step: lay out the pieces from the cardboard, fold them at the look of a rectangle without a lapel, put paper on the arkush and circle the perimeter with an olive. Putting the parts in a different way now, you can tell that the stench, like before, does not go beyond the drawn line, wanting to open the door in the middle of the rectangle.

To our two parts, you can, obviously, add a third, prepared at the sight of smug, like, being applied to one side of the rectangle, turning it into a square; in this rank, we take one more way of cutting a square into three parts, which gives an internal opening.

Curvilinear and trivial options

Pointed by us butts clearly show that the area of ​​paradoxes in the changing area is only beginning to expand. Why do we make like curvilinear figures, for example, a stake or an ellipse, so you can cut them into pieces, and then fold them in such a way, so that when you do this, without a commemorative creation, the figures come out the inside?

Why do we use trivumirnі figures, specific for triokh vimіrіv, then it is not a trivial legacy of two-worldly figures? It’s clear to Adzha that it’s up to a flat figure, with which we were scribbled in this division, you can “add to the world”, vir_zayuchi її just to finish the thick cardboard, the height of such a beautiful “dovzhina of the third world”).

How can a cube or, say, a pyramid be cut apart in a folding way into parts so that, folding them in a new way, take away the memory of the empty ones?

The answer will be like this: if you don’t separate a few parts, then such expanses of figures show it’s not very important. Dosit zrozumilo tse at the time of the cube.

Here empty inner maybe otrimana, prote pitanya about the smallest number of parts, with which one can be reached, folded. Yogo svidomo can be prepared from six parts; Imovirno, which can be reached with a smaller number.

Such a cube can be effectively demonstrated by an offensive rite: swipe yoga from a screen, crushed exactly according to the cube, break it into pieces, revealing a bag in the middle, fold the pieces again into a big cube and show that you are wine (without a bag), like before, box. We are visibly lowered, that we can use a lot of such figures, like flat ones, so spacious, to such a shape that they exude simplicity and thinness. Future successors of the Central Asian region may be satisfied with the results.

butt 1 . For a dart of a 20 cm zavdovka, it is necessary to build a rectangle of the largest area. Know yoga razmіri.

Solution: Significantly one side of the rectangle through x cm, then the other will be (10-x) cm, the area S (x) \u003d (10-x) * x \u003d 10x-x 2;

S/(x)=10-2x; S/(x)=0; x = 5;

For mental tasks x (0;10)

We know the sign similar to the gap (0; 5) and to the gap (5; 10). Pokhіdna change the sign from “+” to “-”. Stars: x = 5 point maximum, S (5) = 25 cm 2 - the most significant. Also, one side of the rectangle is 5cm, the other 10's = 10-5 = 5cm;

butt 2. A plot with an area of ​​2400m 2 needs to be laid out on two straight-cut plots so that the fence is the smallest. Know the difference between the dealers.

Solution: Significantly one side of the plot through x m, then the other will be m, fenced fence P (x) = 3x +;

P / (x) \u003d 3-; P / (x) = 0; 3x 2 = 4800; x 2 \u003d 1600; x=40

For mental tasks x (0; )

We know the sign similar to the gap (0; 40) and to the gap (40; ?). Pokhіdna change the sign from “-” to “+”. Zvіdsi х=40 point minimum, also, P(40)=240m least value, also, one side 40m, the other =60m.

example 3. Dіlyanka of rectangular shape with one side lying down to wake up. When setting a perimeter size of 1 m, it is necessary to fence the plot so that the area is the largest.

Solution:

Significantly, one side of the straight-line plot through x m, then the other will be (-2x) m, area S (x) \u003d (-2x) x \u003d x -2x 2;

S/(x)=-4x; S/(x)=0; -4x; x =;

For mental tasks x (0; )

We know the sign similar to the interval (0; )і to the interval ( ; ). Pokhіdna change the sign from “+” to “-”. Zvіdsi x = point to the maximum. Also, one side of the plot = m, the other -2x = m;

butt 4. From a rectangular sheet of cardboard with sides of 80 cm and 50 cm, it is necessary to make a box of a rectangular shape, turning around the edges of the square and bending the edges, which have settled down. What height can the box be, what is the largest volume?

Solution: Significantly the height of the box (the same side of the curved square) through x m, then one side of the base will be (80-2x) cm, the other (50-2x) cm, volume V (x) \u003d x (80-2x) (50- 2x) \u003d 4x 3 -260x 2 + 4000x;

V / (x) \u003d 12x2 -520x +4000; V/(x)=0; 12x2 -520x +4000 = 0; x 1 = 10; x 2 =

For mental tasks x (0; 25); x 1 (0; 25), x 2 (0; 25)

We know the sign similar to the gap (0; 10) and to the gap (10; 25). Pokhіdna change the sign from “+” to “-”. Zvіdsi x = 10 points to the maximum. Also, box height = 10cm.

Example 5. Dіlyanka of rectangular shape with one side lying down to wake up. When setting the perimeter size of 20 m, it is necessary to enclose the plot so that the area is the largest.

Solution:

Significantly one side of the rectangle through x m, then the other will be (20 -2x) m, area S (x) \u003d (20-2x) x \u003d 20x -2x 2;

S / (x) \u003d 20 -4x; S/(x)=0; 20 -4x = 0; x = = 5;

For mental tasks x (0; 10)

We know the sign similar to the gap (0; 5) and to the gap (5; 10). Pokhіdna change the sign from “+” to “-”. Zvіdsi x = 5 point to the maximum. Also one side of the plot = 5m, the other 20 -2x = 10m;

butt 6 . In order to change the rubbed on the wall and the bottom of the channel, it is necessary to wet the area of ​​​​the area that is possibly small. It is necessary to know the dimensions of a rectilinear canal with a cross section area of ​​4.5 m 2, in which case the area will be the smallest.

Solution:

The depth of the ditch is significant through x m, then the width will be m, P (x) \u003d 2x +;

P / (x) \u003d 2-; P / (x) = 0; 2x 2 \u003d 4.5; x = 1.5. We take less positive significance for the mind task.

For mental tasks x (0; )

We know the sign similar to the gap (0; 1.5) and to the gap (1.5; ?). Pokhіdna change the sign from “-” to “+”. Zvіdsi x=1.5 point to the minimum, also, P(1.5)=6m, the least value, also, one side of the ditch is 1.5m, the other = 3m.

Example 7. Dіlyanka of rectangular shape with one side lying down to wake up. When setting a perimeter size of 200m, it is necessary to enclose the plot so that the area is the largest.

Khrestina Nadiya Mikhaylivna, teacher with children, NOU DOD "DRC "Land of Wonders", Ryazan [email protected]

Entry of TRVZ elements in mathematics lessons

Abstract. At the article, there is a congestion in the lessons of mathematics of the elements of the structure of the creative lesson in the innovative pedagogical system of the NFTMTRIZ. Proposed by the author methodical development the lesson of mathematics in the 5th grade, de demonstrated how it is possible to develop the creativity of students within the framework of the school program. Key words: universal learning activities, creative thought, system-based activity, creative lesson, reflection.

Mathematics is a science, as life is necessary for everyone. From the least age of the child, the light of numbers, forms is thin. І water hour, this world is more foldable and rich in facets. A lot of children, sticking with difficulties at the material, lose interest in the subject and "ignorance" accumulates like a snow sack. That is why the teacher faces a problem: not only to learn, but to add interest, and, therefore, to give the child tools for independent mastering of new knowledge (universal teaching knowledge). myslennya, vminnya pratsyuvati with the problem and virishuvati її, robiti vysnovka, shukati new original approaches, bachit the beauty of the results, what came out. Vіdpovіdalnoї sobistostі uchnya. The standard is dictated to us by the class system of Jan Amos Komensky, in which the teacher is “reply”, and the students are “reply”. New type of lesson, like this: “ brainstorming”, dispute, project activity, to help children in a steadily slow world. What kind of results is the responsibility of the teacher of the results of his work? that self-enlightenment on the basis of motivation until the beginning and recognition of the chosen profession; form communicative competence; put a mark, shlyakhi shlyakhi їх reach, guide the basics of self-control, etc. an increase in the number of winemakers and new professions, a student is guilty, but we are preparing to constantly change drinks for the market. understand that the subject results are now not the only head, it is also necessary to formulate specific metasubjects. The very formulation of the results has changed, the shards of a child can now figure out ways to do that. universal initial activities, like meta-subject results. Only the sukupnіst of universal diy will give the opportunity to formulate in the study vminnya vchitis, like a system. Vaughn gives the opportunity to prostrate at night, as and to what stage those chi and other universal navchalnye are formed. In order to reach the goals, the teacher can help the student with the best elements of the creative pedagogical system of non-stop molding of creative thought (NFTM), in the yakіy є іnstrumenti teorії vіnіshennya vinakhіdnitskih zavdan (TRVZ). , allowing you to learn the lesson is more beautiful, less stressful for the child, keep the child in concentration, everything is busy, and the smut, don’t give him ready knowledge, but give him the opportunity to take them by yourself. Also important nutrition is a private transition from a closed type to a closed type. the type that zachіpayut all the information of the uchnіv, zmushuyut uchіv zamislyuvatisya already for reading the mind, shards є deficient, “remind”, we can avenge too much information. Raznomanіtnіst methodіv prіshennya prizvodnya ruynuvannya psikhologicheskoї іnertsії - zvichki up to standard dіy y znayomіy situatsії аbо pragnennya dіyati vіdpovіdno іn іdpovіdno іn the accumulated knowledge. The collection of possible indications helps to teach the child reflection and self-esteem. novіy vіdmovі vіd closed zavdan. The stench is good in small quantities, if you just need to "get your hand" on a specific formula for power. Ale explaining the new material can be no problem. Aje first food after reading by those at the lesson at the head of the children: “And what’s next?” or “Where do I need it?” All the above has been given to us by the NFTM system - the uninterrupted molding of creative ideas and the development of children's creative abilities. class on the topic “Square of a rectangle. Single squares » Lesson type: Lesson of the development of new material. Objectives for the lesson: 1. Subjects: formulate figures in the text about the area, insert a link between the units of the square, learn from the formulas for the area of ​​a rectangle and a square. Special features: to form in the mind the ways of doing within the framework of the proponated minds and at the same time, adjust your own ideas to the situation that changes. Meta-subjects: to form a mathematical problem in the context of a problematic situation, in an everyday life.

learn to take away the information about the area of ​​\u200b\u200bfigures and її dominance, begin to establish links between units in the world of the area, to set up the formulas for the area of ​​the rectangle and the square; arranging analysis, repairing, sharpening, robiti visnovki; work at the group and couples. Assistant: A.G. Merzlyak, V.B. Polonsky, M.S. Yakir. Mathematics Grade 5 Assistant for uchnіv zagalnosvіtnіh ustanov. 2014 rock.

Etapi to the lesson Task of the stage Teacher's duty School's duty UUD1. Motivation Create a friendly psychological attitude to work, motivation of students to work. Reception, re-reading of preparation before the beginning of employment, organization of respect for children. z'are 8 small ones. - What happened? - What did we do last week? - Today we continue to work with straight-cuts. Include in the business rhythm of the lesson.

Children try to solve the trick. Activate knowledge of the past lesson.

Features: self-appointment. Regulatory: self-organization. Communicative: planning initial interviews with a teacher and classmates. Knowledge: the beginners of the previous work. 2. Zmistovna part. Have sudіv razbrat. The owner of the blue lot, in order to spend on his city, needs to go through the red lot of land. What work? Entrance to the dealerships

To my knowledge, we know that even land plots can be made up of equal areas. What kind of visnovok can we grow? Cholovik vyrishiv get a pidlog at his dacha. Ale pіdloga maє nezvichaynu form. Ale, I don’t know how much farby is needed, on the farboi it says 100g per 1m2. The area of ​​the smaller figure is 12m2, the area of ​​the larger one is 20m2.

Vysuvayut versions in the dispute. Together with the teacher, choose the virna: the blue one needs to take a piece of red earth, and for you, the deputy should be equal in size.

Roaring the visnovok: equal figures make equal squares. development of regulation of primary activity. Communicatives: keep working in the team, just respect the thought of others, keep your position.

Fig. 2Euristic rozmov with elements of the trial and pardon method. On the table at the teacher lie a ruler, a compass, a protractor. We talked about the area, but how can you die? Let's vimiryaєmo square our doshki. What do we have to win the winds? What is for the wimiryuvannya kutiv? Robimo visnovok: for a single vimiryuvannya square, we choose a square, the side of which is equal to a single vіdrіzk. How do we call such a square? In order to minimize the area, it is necessary to improve, how many single squares should be removed from it?

Children sort out all possible tools, come to the visnovka, which is not enough.

-One of the students to go to the board, vvazha, behind the back of the prepared single square with a side of 1 m, the area of ​​the board. topic of the lesson: "Square of a rectangle." 3. Psychological development. Give students the opportunity to change the type of activity. Task for the development of creative zdіbnosti. Orientation in space. 1. A pair of horses ran 20 km. How many kilometers did you run a leather coat? (20 km)2. Klitz had 4 rabbits. Four lads bought one of these rabbits and one rabbit got stuck at the klitz. How could it have gone? (One rabbit was bought at once with a cage) 3. Two hamants have two coins, moreover, in one hamantz there are more coins, lower in the other. How can you booty? (One hamanets lie in the middle of the other) The class is divided into groups of 6 specialties, in the groups a captain is selected as a teacher, after discussing the problem, he chooses the correct answer. 1 whilin is submitted for discussion.

Features: self-appointment. Regulatory: development of regulation of primary activity. Communication: interaction with partners in the field of activity. Knowledge: the beginners of the previous work. The development of a creative mind.

4. Two blues and two fathers laid 3 eggs. How many eggs are z'їv leather? (One egg per skin). Igrashart: “To touch the right ear of the soul is angry with the lіktem of the left hand” 4. Puzzle.

Imagine a system of stacking puzzles embedded in real objects. Self-determination task. 40 km. For how many years you will pass 24 km? 4 with the very speed?

Correct statements.

Fig. 3 In the papers, they write down less than the answers, then they change the papers from the credits according to the part, and they change one into one. In the end, the correct statements are shown on the screen. Features: sensory creation. Regulatory: self-regulation of emotional and functional states, self-organization. The development of a creative mind.

5. Intellectual warm-up. more logical thought and creative abilities. 1. The side of a straight-cut arch of paper has a length of length (in centimeters), and the area of ​​the arch of paper is 12 cm2. How many squares with an area of ​​4 cm2 can be seen from the right rectangle? 2. On the doshtsi through the projector, display an offensive drawing Fig. As with one straight cut, divide the figure into two figures with equal areas. One student at the doshka, others practice from the month. doslіdnitskoї іyalnostі.6.Zmіstovna part.

To replace the program material of the initial course and ensure the formation of systemic ideas and the development of creative zdіbnosti. Is it important for us to respect the area for the help of the square? Thus, one side of the board is 2 m, and the other side is 1 m, the board is rectangular in shape, and can be divided into 2 × 1 single squares. Why do you need more space for doshka? Like a and b - the main sides of the rectangle are expressed in one and the same units. How to know the area of ​​such a rectangle?

Problem. How to know the area of ​​the right chotirikutnik, who has all sides and cuts equal?

Introduced new units of the area: ar (weave), hectare. 1 a = 10 m * 10 m = 100 m2

1 ha = 100 m * 100 m = 10000 m2

For such vimirіv do you need a large single area?

S=ab Learners discuss the problem in groups that were previously formed in psychological research, one group becomes experts (having listened to the hanging versions, they are engaged in their processing and proponuyut one thought to the truth). It is necessary to discuss the solution of the problem. Let's write down the formula for the area of ​​a square S = a 2

-For the protection of the area of ​​land plots, forces, stadiums, etc. Special features: self-appointment. Regulatory: development of regulation of primary activity. . The development of a creative mind.

7. Computer intellectual warm-up. Secure the motivation and the development of the mind. The establishment of correctness and recognition by those.

Computer test. The teacher of control is the number of pardons. fig.5 (the figure is under the table)

Learn to practice on a computer in pairs, take a test. Features: self-appointment. Regulatory: development of regulation of primary activity. Summary. Homework. Presentation of summaries to the lesson. Make sure the return call is on the lesson.

Homework. Denmark's square with a side of 8 cm. Find out the square. Vikoristovuyuchi raznokolorovі shmatochki, explain, and then simplify my hypothesis: 8 * 8 = 65

- The formula for the area of ​​a rectangle, a square, a single square. Budinki uchnі spend dosvіd with parts of the square. Control solution.

Such roses appear, to that between the details, when the rectangle is folded, a gap is established. Features: self-development of moral awareness and orientation of studies in the sphere of moral and ethical knowledge. Regulatory: development of regulation of primary activity.

Letter to the dzherel 1. Federal State Lighting Standard of the Main Global Lighting. federal law RF dated December 17, 2010 No. 1897FZ.2.M.M.Zinovkina. NFTMTRIZ: Creative Lighting of the 21st Century. Moscow, 2007. -313s.

"Zastosuvannya good for the cherry day"

(grade 10)

The methodical system of the teacher's duties to this lesson transfers the molding of the students to independently plan and complete the next work step by step. Learners have the right to consult with the teacher, discuss, and take into account the teacher for the sake of prompting, in a way to help the child learn in different ways, and to recognize the truth.

At the lesson, a discussion of theoretical material is held, the class is divided into groups for the safety of the diversity of the methods they use to propagate with the upcoming selection of the most acceptable ones.

The order of self-sufficient diyalnistyu docily at the lesson of victoriousness to differentiate the task of different equals and to evaluate them accordingly.

An analysis of the results of vikonannya tsikh zavdan uchnyami, krim іnformatsiї pro їkh zavoєnnya, gives the teacher a picture of the main difficulties of the uchnіv, їkh main clearings, which helps to identify the main ways of solving problems.

meta lesson: mastered the mind independently in the complex of developing knowledge, inventing those skills, transferring them to a new mind, vicarious method.

Manager:

Initially-pіznavalna: consolidated, systematization and uzgalnennya knowledge and wisdom, connected with the understanding of the “most and least important functions”; practical zastosuvannya molding reduce that navichok.

Developing: the development of self-study practice, clearly express the thought, conduct self-assessment of the initial activity during the hour of the lesson.

Communication: vminnya take part in the discussion, hear that chuti

Hid lesson

Organizational moment

1. The skin of a person is different in a situation, if you need to know the best way to solve whether it is a task, and mathematics becomes a special case for solving the problems of organization of production, searching for optimal solutions. An important intellectual advancement of the efficiency of production and a reduction in the quality of products and a wide promotion of mathematical methods in technology.

repetition

p align="justify"> The middle task of mathematics is an important role to introduce tasks to extremes, tobto. the task is to know the largest and smallest value, the best, the most profitable, the most economical. With such leaders, Mati to the right to the representations of the RINZNIKH Specialities: INGENERIA-Technologists to be so organized by the Vribnitni, Shcho Vielni Jaknaibi, the constructions want to plan the male on the cosmic ship. sob transport windows were minimal. It can be boldly said that the task of knowing the smallest and maximum values ​​\u200b\u200bis very practical. Today at the lesson we will be busy with the accomplishments of such tasks.

Fixing the woven material

2. Two “stronger” scholars are called up to the doshka (10 min.).

1st student: The Danish tank without a lid looks like a rectangular parallelepiped, which is based on a square and has a volume of 108 cm 3 . For some expansions of the tank for the preparation of the least amount of material?

Solution: Significantly bіk base through x cm, vizimo height of the parallelepiped. We know the bad sign on the proms. Pokhіdna change the sign from “–” to “+”. Zvіdsi х=6 point to the minimum, also, S(6)=108 cm 2 - the least value. Also, the side of the base is 6 divs, the height is 12 divs.

2nd student: In a circle with a radius of 30 cm, there is a rectangle with the largest area. Know yoga razmіri.

Solution: Significantly one side of the rectangle through x cm, then it can be seen the area of ​​the rectangle. We know the sign similar to the gap (0; 30) and to the gap (30; 60). Pokhіdna change the sign from "+" to "-". Zvіdsi х=30 – a point to a maximum. Also, one side of the rectangle is 30, the other side is 30.

3.What time is vipothere is a mutual review on the topic “Sustaining the cold” (1 point is given for the correct skin test). The skin of the student is reconsidered, and for re-verification, she transfers her opinion of the susidov according to the party.

Meals are recorded on a portable doshtsi, it is given less evidence:

The function is called growing for a given interval, so ...

The function is called recessive for a given interval, as ...

The point x 0 is called the minimum point, so ...

The point x 0 is called the maximum point, so ...

Stationary points of the function are called points ...

Write a wild look

Physical changer

Robimo Wisnovki

4. The class is subdivided into groups. Groups win the task of knowing the minimum and maximum of the function.

5. The word is given to "strong" teachers. Learners of the class revise their decisions (XV.).

6. There are tasks for selection for the skin group (10 min.).

1 group.

On the sign "3"

For the function f(х)=х 2 *(6-х) assign the smallest value to the index.

Solution: f (x) \u003d x 2 * (6-x) \u003d 6x 2 + x 3; f / (x) \u003d 12x-3x 2; f/(x)=0; 12x-3x 2 = 0; x 1 = 0; x 2 = 4;

f(0)=0; f(6)=0; f(4)=32-max.

On the sign "4"

For the dart of the zavdovka of 20 cm, you need to build a rectangle of the largest area. Know yoga razmіri.

Solution: Significantly one side of the rectangle through x cm, then the other will be (10-x) cm, area S (x) \u003d (10-x) * x \u003d 10x-x 2; S/(x)=10-2x; S/(x)=0; x = 5. For the mind task x (0; 10). We know the sign similar to the gap (0; 5) and to the gap (5; 10). Pokhіdna change the sign from "+" to "-". Values: х=5 – maximum point, S(5)=25 div 2 – maximum value. Also, one side of the rectangle is 5 cm, the other side is 10x=10-5=5 cm.

On the sign "5"

A plot with an area of ​​2400 m 2 needs to be divided into two plots of a rectangular shape so that the fence was the smallest. Know the difference between the dealers.

Solution: Significantly one side of the plot through x m, let's write down the length of the fence and know better P/(x)=0; 3x2 = 4800; x 2 \u003d 1600; x=40

We know the sign similar to the gap (0; 40) and the gap (40;?). Pokhіdna change the sign from “–” to “+”. Zvіdsi х=40 - minimum point, also, P(40)=240 - least value, also, one side is 40 m, the other - 60 m.

2 group.

On the sign "3"

For the function f(x)=x 2 +(16-x) 2, assign the smallest value to the top.

Solution: f/(x)=2x-2(16-x)x=4x-32; f/(x)=0; 4x-32 = 0; x = 8; f(0)=256; f(16)=256; f(8)=128-min.

On the sign "4"

Dіlyanka of rectangular shape with one side lying down to wake up. When setting the dimensions of the perimeter m, it is necessary to enclose the plot so that the area is the largest.

On the sign "5"

From a rectangular sheet of cardboard with sides of 80 cm and 50 cm, it is necessary to make a box of a rectangular shape, bending along the edges of the square and bending the edges, which are settled. What height can the box be, what is the largest volume?

Significantly the height of the box (the same side of the curved square) through x m, then one side of the base will be (80-2x) cm, the other - (50-2x) cm, volume V (x) \u003d x (80-2x) (50 -2x) \u003d 4x 3, 260x 2 + 4000x; V / (x) \u003d 12x2 -520x +4000; V/(x)=0; 12x2 -520x +4000 = 0.

For mental tasks x (0; 25); x 1 (0; 25), x 2 (0; 25).

We know the sign similar to the gap (0; 10) and to the gap (10; 25). Pokhіdna change the sign from "+" to "-". Zvіdsi х=10 – a point to a maximum. Also, box height = 10 div.

3 group.

On the sign "3"

For the function f(x)=x*(60-x) assign the maximum value per point.

Solution: f(х)=х*(60-х)=60х-х 2; f/(x) = 60-2x; f/(x)=0; 60-2x = 0; x = 30; f(0)=0; f(60)=0; f(30)=900-max.

On the sign "4"

Dіlyanka of rectangular shape with one side lying down to wake up. When setting a perimeter size of 20 m, it is necessary to enclose the plot so that the area is the largest.

Significantly one side of the rectangle through x m, then the other will be (20-2x) m, area S (x) \u003d (20-2x) x \u003d 20x-2x 2; S/(x)=20-4x; S/(x)=0; 20-4x = 0; x \u003d 5. For the mental task x € (0; 10). We know the sign similar to the gap (0; 5) and to the gap (5; 10). Pokhіdna change the sign from "+" to "-". Zvіdsi x=5 - point to the maximum. Also, one side of the plot = 5 m, the other - 20-2 * 5 = 10 m.

On the sign "5"

In order to change the rubbed on the wall and the bottom of the channel, it is necessary to wet the area of ​​​​the area that is possibly small. It is necessary to know the dimensions of a rectilinear canal with a cross section area of ​​4.5 m 2, in which case the area will be the smallest.

Significantly glybin ditch through x m, P/(x) = 0; 2x 2 \u003d 4.5; x = 1.5. We take less positive significance for the mind task. We know the sign similar to the gap (0; 1.5) and to the gap (1.5;?). Pokhіdna change the sign from “–” to “+”. Zvіdsi х=1.5 - minimum point, also, P(1.5)=6 m - least value, also, one side of the ditch - 1.5 m, the other - 3 m.

4 group.

On the sign "3"

For the function f(x)=x 2 (18-x), assign the maximum value per point.

f (x) \u003d x 2 (18-x) \u003d 18x 2 -x 3; f / (x) \u003d (18x 2 -x 3) /; f/(x)=0; 36x-3x 2 = 0; x 1 = 0; x 2 \u003d 12 f (0) \u003d 0; f(18)=0; f(12)=864-max.

On the sign "4".

Dіlyanka of rectangular shape with one side lying down to wake up. When setting the perimeter size of 200 m, it is necessary to enclose the plot so that the area is the largest.

Significantly, one side of a straight-cut plot through x m, then the other will be (200-2x) m, area S (x) = (200-2x) x = 200x-2x 2; S/(x)=200-4x; S/(x)=0; 200-4x = 0; x \u003d 200/4 \u003d 50. For the mental task x (0; 100). We know the sign similar to the gap (0; 50) and to the gap (50; 100). Pokhіdna change the sign from "+" to "-". Zvіdsi х=50 – a point to a maximum. Also, one side of the plot = 50 m, the other - 200-2x = 100 m.

On the sign "5"

It is necessary to prepare a box in the form of a rectangular parallelepiped with a square base, with the smallest volume, so that 300 cm 2 can be prepared for preparation.

Significantly one side of the base through x cm і vizimo volume, then V / (x) \u003d 0 300-3x 2 \u003d 0; x 2 = 100; x \u003d 10. We take more than a positive value for the mind task.

We know the sign similar to the gap (0; 10) and to the gap (10; 0). Pokhіdna change the sign from “–” to “+”. Zvіdsi х=10 - minimum point, also, V(10)=500cm 3 - least value, also, base side - 10 div, height - 50 div.

Asking for class

7. Delegates to the groups explain the decision on the chosen dates (10th century).

8. For the improvement of the balls at the rozmintsі and robots, the badges for the lesson are displayed in the groups.

Give a bag to the lesson

Homework

Razvyazannya tasks for the ball; uchnі, yakі vykonali zavdannya on "5", zvіlnyayutsya in the form of work.

Analysis of the results of vikonannya tsikh zavdan uchnyami, krіm іnformatsії about їх іх іхідідії, giving the teacher a picture of the main difficulties of uchnіv, їх main glades, which will help to identify the main ways of their liquidation.

	FOMKINA TETYANA FEDORIVNA
VISIT CARD
Posada	Teacher of Russian language and literature
Misce roboti	Municipal zahalnosvitnya installation "Serednya skylight school No. 9 "city of Orenburg
Work experience on the estate
Competition ball
The topic of the pedagogical report	Formation of the linguistic competence of students on the basis of a diyalnisno-systemic approach in Russian language teachers for UMK S.I. Lvivsky
sustenity methodical system teacher, who reflects good ideas	The essence of the methodical system of the teacher - in the organization of the initial activity as a turn in the nourishment of the linguistic character (which allows the respect of the students to change the essence of that number of orthographic writing) to the method of diy (on the basis of the rule, the free) operation with the rules in the course of the sheet or vikoristanny of the spelling dictionary.
The work of expanding the power of knowledge, revealing a methodical system of various equals (forms, intellectual products)	Dosvid work Fomkina T.F. zagalneno in 2009 rіvnі MO MOU "ZOSh No. 9" and praised by the methodical council. In 2009 and 2010 pp. representations among teachers of the city of Orenburg on the municipal level. Tetyana Fedorivna spoke at the district methodical public meetings: “Walking ICT for an hour of lessons in Russian language and literature as a form of linguistic competence”, “An active guide to the development of educational standards”.
The effectiveness of the implementation of the methodical system	Formation of stable positive motivation and promotion of the interest of the students to the subject; Positive dynamics of students to the teacher, lessons of Russian literature and literature, development of students' education to predictive activity and activation of knowledge processes; Significantly increase the power creative robots, creative work, which is confirmed by the results of graduation examinations: in 2007, the success rate for the results of the DPA was 100%, the number of assignments for “4” and “5” - 87%; 2008 EDI results success - 100%, the number of students who scored "4" and "5" - 92%, the highest score - 87; in 2009, the success rate for the results of the EDI was 100%; More than a few students, how to take part in scientific and practical conferences, competitions, olympiads: X regional scientific and practical conference of scientists "Ti - Orenburg" (III month), XV month conference of students "Intellectuals of the XXI century" (diploma for "Diversified research of science"), All-Russian correspondence competition "Knowledge and creativity", 2010 (III place, laureate), regional full-time-correspondence competition "Batkivshchyna", 2009 (III month), VI International Olympiad on the Fundamentals of Sciences, 2010 (diplomas of the 1st and 2nd stage), International gra-competition "Russian Vedmezh", 2010 (15 months in the region). Monitoring of public activity shows a high level of education of Fomka Tetyana Fedorivna: Russian language - 69% (2009), literature - 77% (2009).

MATERIALS FOR WORK

A lesson in learning new knowledge

with a different differentiation of education

"NOT with names"

(grade 5)

The presentations of the summary of the lesson are summarized in accordance with the “Programs of Russian language for 5-6 classes” S.I. Lviv (M.; "Mnemosyne", 2008). The lesson is directly shaped by the linguistic, modern and modern competence of the students. The material included in the lesson should be of an initial, developmental nature.

Lesson task:

1) develop communicative thoughts: formulate nutrition and opinions on a grammatical topic; zdiyasnyuvat movna vzaєmodіyu with the mobile group; create authoritative texts on the topic;

2) form linguistic and linguistic competence: know the rule of spelling NOT with a name ; vmіti z supplementary algorithm zastosovuvat tse rule is practical; repeat spelling « NOT with the word " , rule about the name;

3) vikhovuvat dbaylive setting to the word as a spiritual value to the people.

Ownership: multimedia equipment, video presentation, reference cards, test, files from previous tasks.

Hid lesson

Organizational moment

Hello, shovni colleagues! Yes, yes, colleagues. I called you so not vipadkovo. Today we are busy sleeping on the right: virishuvatimemo linguistic tasks, demonstrating the mystery of writing words. Adzhe, for the affirmations of Leo Mikolayovich Tolstoy, “The word is great on the right... With a word you can serve love, with a word you can serve witches and hatred” (Epigraph before the lesson).

Linguistic warm-up "So - ni"

Axis navik volodinnya with a word and help you get into a linguistic warm-up, as it is called “So - nі”. The rules for this warm-up are as follows: I made a rule, and you will try to guess yoga, asking nutrition, what to induce, as if guilty, but formulated in such a way that I could say the words “so” or “ne”. I appreciate your feedback today, I will be tokens. Ask me.

Learn to ask the teacher. For example:

1. Did we start the rule in class 5? (So)

2. What is the rule about the spelling of words? (Hi)

3. Is there a rule about parts of a movie? (So)

4. What is the rule about the name? (So)

- Well done! Guessed!

Actualization of knowledge

And now let's guess what the name is. Ale rozpovіmo about the new one with a lansy, passing one to one the baton, like sportsmen on the wedges. Whoever you want, you can hurry up for the hour help cards. Evaluate your opinions, I will be tokens ( Vіdpovіdі uchnіv).

They screwed up wonderfully! Knowing the rules about names is necessary for us in order to remember to add names to other parts of the movie.

Tse vminnya we perevirimo, vikonavshi Russian rozpodilchiy dictation.

Read the words carefully (by clicking the mouse on the screen of the projector, the image appears).

Ale, what's up? What became of the images? Children, there is a pardon!

Catch її! (Priyom "Catch a pardon")

"Oburyuvatisya" requires writing at once. Why?

Tse dієslovo, yak not get used to without NOT.

(Click on the mouse)

Manager: divide the words into two groups by parts of the mov. (Learn how to win the task)

1. What parts of the promotion did you get? (Names and words)

2. Name names.

3. Name the words.

4. How to spell NOT with the word?

Tsіlepokladannya

Otzhe, knowing the rules about the name and about the spelling NOT with the English help us to get into the new topic, as it sounds like this: "NOT with names".Write її in zoshit.

I wrote down the head of our thoughts in "Dumkovysheet", What is composed of three graphs: “I know”, “I want to know”, “I know (a)”.

In the graph "I know" a rule is given, on the yak mi sogodnі spirits. The whole rule about writing NOT with a word .

In the graph "Wanna Know" the food of the day is formulated: "Z'yasuvati, if it is NOT written with a name at once, but if it is written all at once."

In the graph "Diznavsya" we will write down the food.

Ale back to back vikonaemo vocabulary robot.

Boys, who is it? neviglaі neviglas? What kind of people do we call that? (Vіdpovidі uchnіv)

Write down y zoshit qi words that їх lexical meanings. And now put together with them the phrases of speech (for choice).

Introduction of new material

How do you think, lads, why are the words “neviglas” and “neviglas” written at the same time? (Bo don't get along without NOT)Dopovіd

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