What segments can be drawn to cut. Olympiad, logical and entertaining problems in mathematics

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Experience shows that when using practical teaching methods, it is possible to form in students a number of mental techniques necessary for correctly identifying essential and non-essential features when familiarizing themselves with geometric figures. mathematical intuition, logical and abstract thinking develops, a culture of mathematical speech is formed, mathematical and design abilities are developed, cognitive activity increases, cognitive interest is formed, intellectual and creative potential develops. The article provides a number of practical tasks on cutting geometric shapes into pieces in order to compose these parts create a new figure. Students work on assignments in groups. Then each group defends its project.

Two figures are called equally composed if, by cutting one of them in a certain way into a finite number of parts, it is possible (by arranging these parts differently) to form a second figure from them. So, the partitioning method is based on the fact that any two equally composed polygons are equal in size. It is natural to pose the opposite question: are any two polygons having the same area equal in size? The answer to this question was given (almost simultaneously) by the Hungarian mathematician Farkas Bolyai (1832) and the German officer and mathematics enthusiast Gerwin (1833): two polygons having equal areas are equally proportional.

The Bolyai-Gerwin theorem states that any polygon can be cut into pieces so that the pieces can be formed into a square.

Exercise 1.

Cut the rectangle a X 2a into pieces so that they can be made into a square.

We cut rectangle ABCD into three parts along the lines MD and MC (M is the middle of AB)

Picture 1

We move the triangle AMD so that the vertex M coincides with the vertex C, the leg AM moves to the segment DC. We move the triangle MVS to the left and down so that the leg MV overlaps half of the segment DC. (Picture 1)

Task 2.

Cut the equilateral triangle into pieces so that they can be folded into a square.

Let us denote this regular triangle ABC. It is necessary to cut triangle ABC into polygons so that they can be folded into a square. Then these polygons must have at least one right angle.

Let K be the midpoint of CB, T be the midpoint of AB, choose points M and E on the side AC so that ME=AT=TV=BK=SC= A, AM=EC= A/2.

Figure 2

Let us draw the segment MK and the segments EP and TN perpendicular to it. Let's cut the triangle into pieces along the constructed lines. We rotate the quadrilateral KRES clockwise relative to vertex K so that SC aligns with the segment KV. We rotate the quadrilateral AMNT clockwise relative to the vertex T so that AT aligns with TV. Let's move the triangle MEP so that the result is a square. (Figure 2)

Task 3.

Cut the square into pieces so that two squares can be folded from them.

Let's denote the original square ABCD. Let's mark the midpoints of the sides of the square - points M, N, K, H. Let's draw segments MT, HE, KF and NP - parts of segments MC, HB, KA and ND, respectively.

By cutting the square ABCD along the drawn lines, we obtain the square PTEF and four quadrilaterals MDHT, HCKE, KBNF and NAMP.

Figure 3

PTEF is a ready-made square. From the remaining quadrangles we will form the second square. Vertices A, B, C and D are compatible at one point, segments AM and BC, MD and KS, BN and CH, DH and AN are compatible. Points P, T, E and F will become the vertices of the new square. (Figure 3)

Task 4.

An equilateral triangle and a square are cut out of thick paper. Cut these figures into polygons so that they can be folded into one square, and the parts must completely fill it and must not intersect.

Cut the triangle into pieces and make a square out of them as shown in task 2. Length of the side of the triangle – 2a. Now you should divide the square into polygons so that from these parts and the square that came out of the triangle, you make a new square. Take a square with side 2 A, let's denote it LRSD. Let us draw mutually perpendicular segments UG and VF so that DU=SF=RG=LV. Let's cut the square into quadrangles.

Figure 4

Let's take a square made up of parts of a triangle. Let's lay out the quadrilaterals - parts of the square, as shown in Figure 4.

Task 5.

The cross is made up of five squares: one square in the center, and the other four adjacent to its sides. Cut it into pieces so that you can make a square out of them.

Let's connect the vertices of the squares as shown in Figure 5. Cut off the “outer” triangles and move them to the free spaces inside the ABC square.

Figure 5

Task 6.

Redraw two arbitrary squares into one.

Figure 6 shows how to cut and move the square pieces.

A point is an abstract object that has no measuring characteristics: no height, no length, no radius. Within the scope of the task, only its location is important

The point is indicated by a number or a capital (capital) Latin letter. Several dots - with different numbers or different letters so that they can be distinguished

point A, point B, point C

point 1, point 2, point 3

You can draw three dots “A” on a piece of paper and invite the child to draw a line through the two dots “A”. But how to understand through which ones?

A A A

A line is a set of points. Only the length is measured. It has no width or thickness

Indicated by lowercase (small) Latin letters

a b c

1. The line may be
2. closed if its beginning and end are at the same point,

open if its beginning and end are not connected

closed lines

1. You left the apartment, bought bread at the store and returned back to the apartment. What line did you get? That's right, closed. You are back to your starting point. You left the apartment, bought bread at the store, went into the entrance and started talking with your neighbor. What line did you get? Open. You haven't returned to your starting point. You left the apartment and bought bread at the store. What line did you get? Open. You haven't returned to your starting point.
2. self-intersecting

without self-intersections

self-intersecting lines

1. lines without self-intersections
2. straight
3. broken

crooked

straight lines

broken lines

curved lines

A straight line is a line that is not curved, has neither beginning nor end, it can be continued endlessly in both directions

Even when a small section of a straight line is visible, it is assumed that it continues indefinitely in both directions

Indicated by a lowercase (small) Latin letter. Or two capital (capital) Latin letters - points lying on a straight line

a

B A

1. Direct may be
   • intersecting if they have a common point. Two lines can intersect only at one point.
2. perpendicular if they intersect at right angles (90°).

Parallel, if they do not intersect, do not have a common point.

parallel lines

intersecting lines

A ray is a part of a straight line that has a beginning but no end; it can be continued indefinitely in only one direction

The ray of light in the picture has its starting point as the sun.

Sun

A point divides a straight line into two parts - two rays A A

The beam is designated by a lowercase (small) Latin letter. Or two capital (capital) Latin letters, where the first is the point from which the ray begins, and the second is the point lying on the ray

ray a

beam AB

The rays coincide if

1. located on the same straight line
2. start at one point
3. directed in one direction

rays AB and AC coincide

rays CB and CA coincide

A segment is a part of a line that is limited by two points, that is, it has both a beginning and an end, which means its length can be measured. The length of a segment is the distance between its starting and ending points

Through one point you can draw any number of lines, including straight lines

Through two points - an unlimited number of curves, but only one straight line

curved lines passing through two points

a

A piece was “cut off” from the straight line and a segment remained. From the example above you can see that its length is the shortest distance between two points.

✂ B A ✂

A segment is denoted by two capital (capital) Latin letters, where the first is the point at which the segment begins, and the second is the point at which the segment ends

segment AB

Problem: where is the line, ray, segment, curve?

A broken line is a line consisting of consecutively connected segments not at an angle of 180°

A long segment was “broken” into several short ones

The links of a broken line (similar to the links of a chain) are the segments that make up the broken line. Adjacent links are links in which the end of one link is the beginning of another. Adjacent links should not lie on the same straight line.

The vertices of a broken line (similar to the tops of mountains) are the point from which the broken line begins, the points at which the segments that form the broken line are connected, and the point at which the broken line ends.

A broken line is designated by listing all its vertices.

broken line ABCDE

vertex of polyline A, vertex of polyline B, vertex of polyline C, vertex of polyline D, vertex of polyline E

broken link AB, broken link BC, broken link CD, broken link DE

link AB and link BC are adjacent

link BC and link CD are adjacent

A B C D E 64 62 127 52

The length of a broken line is the sum of the lengths of its links: ABCDE = AB + BC + CD + DE = 64 + 62 + 127 + 52 = 305 Task: which broken line is longer , A? The first line has all the links of the same length, namely 13 cm. The second line has all the links of the same length, namely 49 cm. The third line has all links of the same length, namely 41 cm.

A polygon is a closed polyline

The sides of the polygon (the expressions will help you remember: “go in all four directions”, “run towards the house”, “which side of the table will you sit on?”) are the links of a broken line. Adjacent sides of a polygon are adjacent links of a broken line.

The vertices of a polygon are the vertices of a broken line. Adjacent vertices are the endpoints of one side of the polygon.

A polygon is denoted by listing all its vertices.

closed polyline without self-intersection, ABCDEF

polygon ABCDEF

polygon vertex A, polygon vertex B, polygon vertex C, polygon vertex D, polygon vertex E, polygon vertex F

vertex A and vertex B are adjacent

vertex B and vertex C are adjacent

vertex C and vertex D are adjacent

vertex D and vertex E are adjacent

vertex E and vertex F are adjacent

vertex F and vertex A are adjacent

polygon side AB, polygon side BC, polygon side CD, polygon side DE, polygon side EF

side AB and side BC are adjacent

side BC and side CD are adjacent

CD side and DE side are adjacent

side DE and side EF are adjacent

side EF and side FA are adjacent

The perimeter of a polygon is the length of the broken line: P = AB + BC + CD + DE + EF + FA = 120 + 60 + 58 + 122 + 98 + 141 = 599

A polygon with three vertices is called a triangle, with four - a quadrilateral, with five - a pentagon, etc.

A series of elective classes on the topic “Solving cutting problems”

Explanatory note

Basic goals that we put in elective classes are as follows:

Present material about the types of cutting polygons;

To promote the formation of skills in students to mentally carry out such transformations as:

• parallel transfer,

  turn,

  central symmetry and various compositions of these transformations.

AND the main goal of all classes: achieve a positive change in spatial thinking abilities.

The tasks offered in elective classes are creative in nature, their solution requires students to: skills:

the ability to make mental transformations that modify the location of the images students have in their minds, their structure, structure;

the ability to change the image both in location and structure simultaneously and repeatedly perform compositions of individual operations.

Thematic planning:

1. Questionnaire No. 1 – 1 hour.

2. Cutting problems. Type R cutting – 1 hour.

3. Type P cutting – 1 hour.

4. Q type cutting – 1 hour.

5. Type S cutting – 1 hour.

6. T-type cutting – 1 hour.

7. Questionnaire No. 2 – 1 hour.

When compiling a series of elective classes, problems from the magazines “Kvant”, “Mathematics at School” and the book by G. Lindgren were used.

Guidelines: When introducing students to problems, we recommend considering these problems precisely according to the types of cutting proposed by G. Lindgren, which allows, on the one hand, to classify these problems, on the other hand, in the classroom to solve problems involving spatial transformations of various levels of complexity (the second and third types operating with images, according to I.S. Yakimanskaya). We recommend using the tasks of elective classes when working with students in grades 7–9.

Lesson No. 1

Topic: Cutting problems. Cutting type R (rational cutting).

Target: To acquaint students with the concept of a cutting problem, explain the essence of cutting type R, analyzing the solution of problems for this type of cutting, in the process of solving problems, promote the formation of skills to mentally carry out operations (cutting, adding, re-cutting, turning, parallel transfer), thereby promoting the development spatial thinking.

Equipment: paper, colored pastes, scissors, poster.

Method: explanatory - illustrative.

Teacher: poster on the board:

Scheme: Cutting problems

Cutting problems

1) Cut the figure into several figures

3) Reshape one or more shapes into another shape

2) Fold a figure from the given figures

Among all cutting problems, most of them are rational cutting problems. This is due to the fact that such cuts are easy to come up with and the puzzles based on them are not too simple and not too complex.

Problems in R - cutting

1) Cut the figure into several (mostly equal) figures

3) Reshape one or more shapes into a given shape

2) Add a figure from given (mostly equal) figures

3.1. Using step cutting

3.2. Without using step cutting

Let's get acquainted with the solution of problems for each type of cutting R.

Stage II: Problem solving stage

Methods: partial-search

Task No. 1(AII) : Cut a square with a side of four squares into two equal parts. Find as many ways to cut as possible.

Note: You can only cut along the sides of the cells.

Solution:

Students search for such cuts in their notebooks, then the teacher summarizes all the cutting methods found by the students.

Problem No. 2(AII) : Cut these shapes into two equal parts.

Note: You can cut not only along the sides of the cells, but also diagonally.

Students search for such cuts in their notebooks with the help of the teacher.

The square has many wonderful properties. Right angles, equal sides, symmetry give it simplicity and perfection of form. There are many puzzles on folding squares from parts of the same and different shapes.

TO example task No. 3(BII) : You are given four identical parts. Make a square out of them mentally, using all four parts each time. Do all tests on paper. Present the results of your solution in the form of a hand-drawn drawing.

Solution:

A chessboard cut into pieces, which must be folded correctly, is one of the popular and well-known puzzles. The complexity of the assembly depends on how many parts the board is divided into.

I propose the following task:

Problem No. 4(BII) : Assemble a chessboard from the parts shown in the picture.

Solution:

Problem #5(VII) : Cut the “Boat” into two parts so that you can fold them into a square.

Solution:

1) cut into two parts as in the picture

turn one of the parts over (i.e., rotate)

Problem No. 6(VII): Any of the three figures can be cut into two parts, from which it is easy to fold a square. Find such cuts.

A) b)

V)

Solution:

parallel transfer of part 1 relative to part 2

rotation of part 1 relative to part 2

) b) V)

Problem No. 7(VII): A rectangle with sides of 4 and 9 units is cut into two equal parts, which, when folded properly, could be obtained as a square.

the cut is made in the form of steps, the height and width of which are the same;

the figure is divided into parts and one part is moved up one (or several) steps, placing it on another part.

Solution:

parallel transfer of part 1

Problem No. 9(VII): Having cut the figure shown in the figure into two parts, fold them into a square so that the colored squares are symmetrical with respect to all axes of symmetry of the square.

Solution:

parallel transfer of part 1

Problem No. 9(ВIII): How should two squares 3 x 3 and 4 x 4 be cut so that the resulting parts can be folded into one square? Come up with several ways. Try to get by with as few parts as possible.

Solution:

parallel transfer of parts

Way:

Way:

parallel translation and rotation

4 way:

parallel transfer and rotation of parts

Students, with the help of the teacher, search for cuts.

Problem No. 10(AIII): The figure shown in the figure must be divided into 6 equal parts, making cuts only along the grid lines. In how many ways can you do this?

Solution: Two possible solutions.

Problem No. 11(BII): Build a chessboard from the given pieces.

Solution:

Problem No. 12(BIII): Convert the 3 x 5 rectangle into a 5 x 3 rectangle without rotating the corresponding parts.

Note: Use step cutting.

Solution:(parallel transfer)

Problem No. 13(BIII): Cut the shape into 2 pieces with one cut to form an 8 x 8 square.

Solution:

rotation of part 2 relative to part 1

Guidelines: Type R cutting problems are some of the easiest and most interesting. Many problems for this type of cutting involve several methods of solution, and students’ independent solution of these problems can help identify all the methods of solution. Tasks 1, 2, 3, 6, 7, 8, 10, 12, 13 involve students working with the image of figures, through mental transformations (“cutting”, addition, rotation, parallel transfer). Problems 4, 5, 9, 11 involve students working with models (made of paper), by directly cutting the figure with scissors and carrying out mathematical transformations (rotation, parallel translation) to find solutions to the problems. Tasks 1, 2, 3, 4, 5, 6, 7, 8, 11, 13 - for the second type of operating with images, tasks 9, 10, 12 - for the third type of operating with images.

Lesson No. 2

Topic: Cutting type P (P parallelogram shift).

Target: Explain the essence of cutting type P, in the process of analyzing the solution of problems for this type of cutting, while promoting the formation of skills to mentally carry out operations (cutting, adding, re-cutting, parallel transfer), thereby promoting the development of spatial thinking.

Equipment:

Stage I: Orientation stage

Method: problematic presentation.

Teacher poses a problem (solve problem No. 1) and shows its solution.

Task No. 1(BIII): Convert a parallelogram with sides of 3 and 5 cm into a new parallelogram with the same angles as the original parallelogram, one of whose sides is 4 cm.

Solution: 1)

4)

ABC D – parallelogram

AB = 3, A D=5

make a cut AO VO = D K = 4;

move part 1 up (parallel translation) to the right along the cut line until point O falls on the continuation of side DC;

make a cut KA' so that KA' || DC ;

and Δ AA'K we insert into the recess located below point O (parallel transfer of Δ AA'K along straight line AO).

KVO D is the desired parallelogram (КD = 4)

KDO=  A.D.C.  BAD =  1 +  4,

1 =  2 and  4 =  3 – lying crosswise on parallel lines.

Therefore,  BAD =  2 + 3 =  BOC =  BKD,  BAD =  BKD, etc.

U

Problems on P shift

Reshape one or more shapes into another shape

The essence of cutting type P:

we make a section of this figure that meets the requirements of the task;

we carry out a parallel transfer of the cut part along the cut line until the top of the cut part coincides with the continuation of the other side of the original figure (parallelogram);

make a second cut parallel to the side of the parallelogram, we get another part;

We carry out a parallel transfer of the newly cut part along the line of the first cut until the vertices coincide (we put the part into the recess).

Stage II: Problem solving stage

Methods: explanatory - illustrative

Problem No. 2(BII): Convert the 5 x 5 square into a rectangle with a width of 3.

Solution:

1) 2) – 3) 4)

section AO / VO = D T = 3

parallel transfer ΔABO along straight line AO ​​until point O  (DC)

cut TA’ / TA’ || CD

Δ AA ’T parallel transfer along straight line AO.

TBOD is the desired rectangle (TB = 3).

Problem No. 3(ВIII): Fold three identical squares into one large square.

Note: Fold three squares into a rectangle then apply P shift.

Solution:

S pr = 1.5 * 4.5 = 6.75

kv = 6.75 =

1) 2) – 3)

4)

Problem No. 4(BIII): Cut the 5 x 1 rectangle into a square

Note: make an incision AB (A W =
), apply P shift to the rectangle XYWA.

Solution:

1)

2) – 3) 4) 5)

Problem No. 5(ВIII): Convert the Russian Н into a square.

Note: make a cut as shown in the figure, fold the resulting parts into a rectangle.

Solution:

Problem No. 6(BIII): Convert the triangle into a trapezoid.

Note: Make the cut as shown in the picture.

Solution:

rotate part 1;

AB section;

ΔАВС parallel transfer along AB until point B  (FM)

cut OR / OR || FM;

ΔAOR by parallel transport along AB. Point P coincides with point B;

OFBC is the desired trapezoid.

Problem No. 7(ВIII): Make one square from three equal Greek crosses.

Solution:

Problem No. 8(BIII): Convert the letter T into a square.

Note: First, cut out a rectangle from the letter t.

Solution: S t = 6 (unit 2), Skv = (
) 2

turn

composition of parallel hyphens

MV = KS =

Problem No. 9(ВIII): Redraw the flag shown in the picture into a square.

Note: First convert the flag to a rectangle

Solution:

turn

S fl = 6.75 AB = C D =
Skv = (
) 2

parallel transfer

Guidelines: When introducing students to type P cutting problems, we recommend that they present the essence of this type of cutting when solving a specific problem. We recommend solving problems first on models (made of paper), by directly cutting the figures with scissors and performing parallel transfer, and then, in the process of solving problems, from models of figures to moving on to working with images of geometric shapes, by carrying out mental transformations (cutting, parallel transfer).

Lesson No. 3

Topic: Cutting type Q (Q is a shift of a quadrilateral).

Target: Let us outline the essence of cutting type Q, in the process of solving problems for this type of cutting, while promoting the formation of skills to mentally carry out operations (cutting, addition, central symmetry, rotation, parallel transfer), thereby promoting the development of spatial thinking.

Equipment: paper, colored pastes, scissors.

Stage I: Orientation stage

Method: problematic presentation.

The teacher poses a problem to the students (solve problem No. 1) and shows the solution.

Task No. 1(BIII): Convert this quadrilateral into a new quadrilateral.

Solution:

We make the HP cut so that VN = MN, PF = DF;

make a cut ME / ME || Sun;

we carry out the RT / RT incision || AD ;

Δ 3 and Δ 1 are rotated clockwise relative to part 2;

Part 1 by parallel transfer along a straight line HF until point T  AR;

AMCP is the required quadrilateral (with sides CP and AM (can be specified in the condition)).

Problem No. 2(BIII): Convert the quadrilateral into a new quadrilateral (long quadrilateral).

Solution:

(rotate part 1 relative to point O until OU coincides with AO);

(rotate part (1 – 2) relative to point T until VT coincides with WT);

XAZW is the required quadrilateral.

In problems using Q cuts, cuts are made and the cut pieces undergo a rotation transformation.

Tasks for Q cutting

transform a given shape (quadrangle) into another shape (quadrangle)

In many problems, Q shift elements are used to transform a triangle into some kind of quadrilateral or vice versa (a triangle as a "quadrilateral" with one of its sides having zero length).

Stage II: Problem solving stage

Problem No. 3(VII): A small triangle is cut from the triangle, as shown in the figure. Rearrange the small triangle to form a parallelogram.

Rotate part 1 relative to point P until KR coincides with MR.

AOO'M is the required parallelogram.

Problem No. 4(BII, BIII): Which of these triangles can be turned into rectangles by making one (two) cuts and rearranging the resulting parts?

1) 2) 3) 4)

5)

Solution:

1)

5)

1), 5) one cut (cut – the middle line of the triangle)

2)

3)

4)

2), 3), 4) two cuts (1st cut – midline, 2nd cut – height from the vertex of the triangle).

Problem No. 5(VII): Rebuild the trapezoid into a triangle.

Solution:

section KS (AK = KB)

rotation ΔKVS around point K so that the segments KV and KA are aligned.

Δ FCD the desired triangle.

Problem No. 6(ВIII): How to break a trapezoid into shapes from which you can make a rectangle?

Solution:

1) OR section (AO = OB, OR┴AD)

2) cut TF (CT = TD, TF ┴AD)

rotation of part 1 relative to point O so that AO and BO are aligned.

Rotate part 2 relative to point T so that DT and CT are aligned.

PLMF – rectangle.

Stage III: setting homework.

Problem No. 7(ВIII) : convert any triangle to a right triangle.

Comment:

1) first convert an arbitrary triangle to a rectangle.

2) rectangle into right triangle.

Solution:

turn

Problem No. 8(VII): Convert an arbitrary parallelogram into a triangle by making only one cut.

Solution:

turn

Rotate part 2 around point O by 180º (center of symmetry)

Guidelines: Summary of the essence of Q cutting we recommend

carry out in the process of solving specific problems. The main mathematical transformations used in solving problems for this type of cutting are: rotation (in particular, central symmetry, parallel translation). Tasks 1, 2, 7 – for practical actions with models of geometric shapes; tasks 3, 4, 5, 6, 8 involve working with images of geometric shapes. Tasks 3, 4, 5, 8 – for the second type of operating with images, tasks 1, 2, 4, 6, 7 – for the third type of operating with images.

Lesson No. 4.

Topic: Type S cutting.

Target: Explain the essence of cutting type S, in the process of solving problems for this type of cutting, while promoting the formation of skills to mentally carry out operations (cutting, adding, overlapping, turning, parallel transfer, central symmetry), thereby promoting the development of spatial thinking.

Equipment: paper, colored pastes, scissors, code positives.

I stage: Oriented stage.

Method: explanatory and illustrative.

Task No. 1(VII): how to cut a parallelogram, whose sides are 3.5 cm and 5 cm, into a parallelogram with sides 3.5 cm and 5.5 cm, making only one “cut”?

Solution:

1) draw a segment (cut) CO = 5.5 cm, divide the parallelogram into two parts.

2) we apply the triangle COM to the opposite side of the parallelogram AK. (i.e. parallel transfer of ∆ COM to the segment SA in the direction of SA).

3) CAOO` is the desired parallelogram (CO = 5.5 cm, CA = 3.5 cm).

Task No. 1(ВIII): show how you can cut a square into 3 parts so that you can use them to make a rectangle with one side twice the size of the other.

Solution:

Construct square ABCD

let's draw the diagonal AC

Let's draw half of the diagonal BD segment OD (OD ┴AC), OD = ½ AC. Build a rectangle from the resulting 3 parts (length AC, width AD

For this:

carry out a parallel transfer of parts 1 and 2. part 1 (∆1) in the direction D A, ∆2 in the direction AB to segment AB.

АОО`С is the required rectangle (with sides AC, OA = ½ AC).

Teacher: We have considered the solution of 2 problems; the type of cutting used in solving these problems is figuratively called S-cutting.

S -cutting is basically the transformation of one parallelogram into another parallelogram.

The essence of this cut in the following:

we make a cut equal in length to the side of the required parallelogram;

we carry out a parallel transfer of the cut part until the equal opposite sides of the parallelogram coincide (i.e. we apply the cut part to the opposite side of the parallelogram)

Depending on the requirements of the task, the number of cuts will depend.

Let's consider the following tasks:

Task No. 3(BII): divide the parallelogram into two parts from which you can add a rectangle.

Let's draw an arbitrary parallelogram.

Solution:

from point B, lower the height of VN (VN┴AD)

Let us carry out a parallel transfer of ∆ AVN to the segment BC in the direction of BC.

Draw a drawing of the resulting rectangle.

VNRS – rectangle.

Task No. 4(BIII): The sides of the parallelogram are 3 and 4 cm. Turn it into a parallelogram with sides of 3.5cm by making two cuts.

Solution:

1)

2)

The desired parallelogram.

In general, S-cutting is based on the method of superimposing strips, which allow solving the problem of transforming any polygons.

In the above problems, due to their ease, we dispensed with the method of applying stripes, although all these solutions can be obtained using this method. But in more complex tasks you cannot do without stripes.

Briefly stripe method boils down to this:

1) Cut (if necessary) each polygon (the polygon that is being transformed and the polygon into which the original polygon must be transformed) into parts from which two strips can be folded.

2) Place the strips on top of each other at a suitable angle, with the edges of one of them always positioned equally in relation to the elements of the other strip.

3) In this case, all the lines located in the common part of the 2 strips will show the places of the necessary cuts.

Letter S, used in the term “S-cut”, comes from the English Strip - strip.

Stage II: Problem solving stage

Using problem 3 as an example, let us verify that the method of applying stripes gives the desired solution.

Problem No. 3(VII): Divide the parallelogram into two parts from which you can add a rectangle.

Solution:

1)

2)

3)

1) we get a strip from a parallelogram

2) stripes of rectangles

3) apply strip 2 to strip 1, as shown in Figure 3

4) we obtain the required task.

Problem No. 5(BIII): In an isosceles triangle, the midpoints of the lateral sides and their projections onto the base are marked. Two straight lines are drawn through the marked points. Show that the resulting pieces can be used to form a rhombus.

Solution:

part 2, 3 – rotation around a point

part 4 – parallel transfer

In this problem, the cutting of triangles has already been indicated; we can verify that this is an S-cut.

Problem No. 6(BIII): Convert three Greek crosses into a square (using stripes).

Solution:

1)

We put a strip of squares on a strip of crosses so that point A and point C belong to the edges of the strip of crosses.

∆АВН = ∆СD B, therefore, the square consists of ∆АВС and ∆АВМ.

Stage III: Setting homework

Problem No. 7(BIII): Convert this rectangle into another rectangle, the sides of which are different from the sides of the original rectangle.

Note: Look at the solution to problem 4.

Solution:

section AO (AO – width of the required rectangle);

cut DP / DP  AO (DP – length of the required rectangle);

parallel transfer of ∆AVO in the direction of the aircraft to the segment of the aircraft;

parallel transfer of ∆АPD to the segment AO in the direction of AO;

PFED required rectangle.

Problem No. 8(BIII): A regular triangle is divided into parts by a segment; from these parts, fold a square.

Note: You can verify by overlaying the strips that this is an S cut.

rotation of part 2 around point O;

rotation of part 3 around point C;

parallel transfer of part 4

Additional task No. 9(BII): Cut the parallelogram along a straight line passing through its center, so that the resulting two pieces can be folded into a rhombus.

Solution:

O  QT

QT cut;

part 1 by parallel transfer to the BC segment in the direction BC (CD and AB are combined).

Guidelines: S – cutting – one of the most difficult types of cutting. We recommend that the essence of this cutting be outlined in specific tasks. In classes on solving problems on S - cutting, we recommend using problems in which cutting figures are given and it is necessary to add the required figure from the resulting parts, this is explained by the difficulty of students independently implementing the method of applying strips, which is the essence of S - cutting. At the same time, on tasks that are more accessible to students (for example, on tasks 3, 5, 8), the teacher can show how the method of applying strips allows one to obtain the cuts given in the task conditions. Tasks 4, 5, 6, 8, 9 – for practical actions with models of geometric shapes, tasks 1, 2, 3, 7 – for working with images of geometric shapes. Tasks 1, 3, 9 – for the second type of operating with images, tasks 2, 4, 5, 6, 7, 8 – for the third type of operating with images.

Lesson No. 5

Topic: T-type cutting.

Target: Explain the essence of cutting type S, in the process of analyzing the solution of problems for this type of cutting, while promoting the formation of skills to mentally carry out operations (cutting, adding, turning, parallel transfer), thereby promoting the development of spatial thinking.

Equipment: paper, colored pastes, scissors, colored pastes, code positives.

Stage I: Orientation stage

Method: explanatory and illustrative

Teacher: Using T-cutting to solve problems involves drawing up a mosaic and their subsequent overlay. The strips used in S-cutting can be obtained from mosaics. Therefore, the tiling method generalizes the strip method.

Let's look at the essence of T-cutting using the example of problem solving.

Task No. 1(BIII): Convert the Greek cross into a square.

1) the first step is to convert the original polygon into a mosaic element (and this is necessary);

2) from these elements we make mosaic No. 1 (we make a mosaic from Greek crosses);

5) all lines located in the common part of the two mosaics will show the places of the necessary cuts.

Stage II: Problem solving stage

Method: partially - search

Problem No. 2(BIII): The Greek cross is cut into three parts, fold these parts into a rectangle.

Note: we can verify that this cut is a T-type cut.

Solution:

rotation of part 1 around point O;

rotate part 2 around point A.

Problem No. 3(BIII): Cut the convex quadrilateral along two straight lines connecting the midpoints of opposite sides. Show that from the resulting four pieces it is always possible to add a parallelogram.

part 2 rotation around point O (or center of symmetry) by 180;

part 3 rotation around point C (or center of symmetry) by 180;

part 1 – parallel transfer.

Let us show the mosaic from which this cut was obtained.

Problem No. 4(BIII): Three identical triangles were cut along different medians. Fold the six resulting pieces into one triangle.

Solution:

1) from these triangles we make triangles as in Figure 1 (central symmetry);

2) we make another triangle from three new triangles (equal sides coincide).

Let's show how these sections were made using mosaics.

Problem No. 5(BIII): The Greek cross was cut into pieces, and a right-angled isosceles triangle was made from these pieces.

Solution:

part 1 central symmetry;

part 3 central symmetry;

parts 3 and 4 – turn.

Problem No. 6(BIII): Cut this figure into a square.

Solution:

part 1 rotation around point O;

part 3 turn 90 around point A.

Problem No. 7(BIII): Cut the Greek cross into a parallelogram (cuts are given).

Solution:

part 2 – parallel transfer relative to part 1;

part 3 parallel transfer along the cut line.

Stage III: Setting homework.

Problem No. 8(BIII): Two identical paper convex quadrangles with cuts: the first along one of the diagonals, and the second along the other diagonal. Prove that the resulting parts can be used to form a parallelogram.

Solution: composition of turns.

Problem No. 9(BIII): Make a square from two identical Greek crosses.

Solution:

Guidelines: T - cutting - the most complex type of cutting, forming cuts of type S. We recommend that you explain the essence of T-cutting in the process of solving problems. Due to the complexity of implementing the mosaic method for students, which is the essence of T-cutting, in the classroom we recommend using tasks in which cutting is specified and it is required to obtain the desired figure from the resulting parts of the figure using mathematical transformations (rotation, parallel translation). At the same time, on tasks that are more accessible to students, the teacher can show how to obtain cutting data using the mosaic method. The tasks proposed in lesson No. 5 are for the third type of operating with images and involve students working with models of geometric figures by performing rotation and parallel translation.

Teacher's introduction:

A little historical background: Many scientists have been interested in cutting problems since ancient times. Solutions to many simple cutting problems were found by the ancient Greeks and Chinese, but the first systematic treatise on this topic was written by Abul-Vef. Geometers began seriously solving problems of cutting figures into the smallest number of parts and then constructing another figure in the early 20th century. One of the founders of this section was the famous puzzle founder Henry E. Dudeney.

Nowadays, puzzle lovers are keen on solving cutting problems because there is no universal method for solving such problems, and everyone who undertakes to solve them can fully demonstrate their ingenuity, intuition and ability for creative thinking. (During the lesson we will indicate only one of the possible examples of cutting. It can be assumed that students may end up with some other correct combination - there is no need to be afraid of this).

This lesson is supposed to be conducted in the form of a practical lesson. Divide the circle participants into groups of 2-3 people. Provide each group with figures prepared in advance by the teacher. Students have a ruler (with divisions), a pencil, and scissors. It is allowed to make only straight cuts using scissors. Having cut a figure into pieces, you need to make another figure from the same parts.

Cutting tasks:

1). Try cutting the figure shown in the figure into 3 equal-shaped parts:

Hint: The small shapes look a lot like the letter T.

2). Now cut this figure into 4 equal-shaped parts:

Hint: It is easy to guess that small figures will consist of 3 cells, but there are not many figures with three cells. There are only two types: corner and rectangle.

3). Divide the figure into two equal parts, and use the resulting parts to form a chessboard.

Hint: Suggest starting the task from the second part, as if getting a chessboard. Remember what shape a chessboard has (square). Count the available number of cells in length and width. (Remember that there should be 8 cells).

4). Try cutting the cheese into eight equal pieces with three movements of the knife.

Tip: try cutting the cheese lengthwise.

Tasks for independent solution:

1). Cut out a square of paper and do the following:

· cut into 4 pieces that can be used to make two equal smaller squares.

· cut into five parts - four isosceles triangles and one square - and fold them so that you get three squares.

For the attention of mathematics tutors and teachers of various electives and clubs, a selection of entertaining and educational geometric cutting problems is offered. The goal of a tutor using such problems in his classes is not only to interest the student in interesting and effective combinations of cells and figures, but also to develop his sense of lines, angles and shapes. The set of problems is aimed mainly at children in grades 4-6, although it is possible to use it even with high school students. The exercises require students to have a high and stable concentration of attention and are perfect for developing and training visual memory. Recommended for mathematics tutors preparing students for entrance exams to mathematics schools and classes that place special demands on the child’s level of independent thinking and creative abilities. The level of tasks corresponds to the level of entrance Olympiads to the Lyceum “second school” (second mathematical school), the Small Faculty of Mechanics and Mathematics of Moscow State University, the Kurchatov School, etc.

Math Tutor Note:
In some solutions to problems, which you can view by clicking on the corresponding pointer, only one of the possible examples of cutting is indicated. I fully admit that you may end up with some other correct combination - no need to be afraid of that. Check your little one's solution carefully and if it satisfies the conditions, then feel free to take on the next task.

1) Try cutting the figure shown in the figure into 3 equal-shaped parts:

: Small shapes are very similar to the letter T

2) Now cut this figure into 4 equal-shaped parts:

Math tutor tip: It’s easy to guess that small figures will consist of 3 cells, but there are not many figures with three cells. There are only two types of them: a corner and a 1×3 rectangle.

3) Cut this figure into 5 equal-shaped pieces:

Find the number of cells that make up each such figure. These figures look like the letter G.

4) Now you need to cut a figure of ten cells into 4 unequal rectangle (or square) to each other.

Math tutor instructions: Select a rectangle, and then try to fit three more into the remaining cells. If it doesn't work, change the first rectangle and try again.

5) The task becomes more complicated: you need to cut the figure into 4 different in shape figures (not necessarily rectangles).

Math tutor tip: first draw separately all types of figures of different shapes (there will be more than four of them) and repeat the method of enumerating options as in the previous task.
:

6) Cut this figure into 5 figures from four cells of different shapes so that only one green cell is painted in each of them.

Math tutor tip: Try to start cutting from the top edge of this figure and you will immediately understand how to proceed.
:

7) Based on the previous task. Find how many figures of different shapes there are, consisting of exactly four cells? The figures can be twisted and turned, but you cannot lift the table (from its surface) on which it lies. That is, the two given figures will not be considered equal, since they cannot be obtained from each other by rotation.

Math tutor tip: Study the solution to the previous problem and try to imagine the different positions of these figures when turning. It is not difficult to guess that the answer to our problem will be the number 5 or more. (In fact, even more than six). There are 7 types of figures described.

8) Cut a square of 16 cells into 4 equal-shaped pieces so that each of the four pieces contains exactly one green cell.

Math tutor tip: The appearance of the small figures is not a square or a rectangle, or even a corner of four cells. So what shapes should you try to cut into?

9) Cut the depicted figure into two parts so that the resulting parts can be folded into a square.

Math tutor hint: There are 16 cells in total, which means the square will be 4x4 in size. And somehow you need to fill the window in the middle. How to do it? Could there be some kind of shift? Then, since the length of the rectangle is equal to an odd number of cells, the cutting should be done not with a vertical cut, but along a broken line. So that the upper part is cut off on one side of the middle cell, and the lower part on the other.

10) Cut a 4x9 rectangle into two pieces so that they can be folded into a square.

Math tutor tip: There are 36 cells in total in the rectangle. Therefore, the square will be 6x6 in size. Since the long side consists of nine cells, three of them need to be cut off. How will this cut proceed?

11) The cross of five cells shown in the figure needs to be cut (you can cut the cells themselves) into pieces from which a square could be folded.

Math tutor tip: It is clear that no matter how we cut along the lines of the cells, we will not get a square, since there are only 5 cells. This is the only task in which cutting is allowed not by cells. However, it would still be good to leave them as a guide. for example, it's worth noting that we somehow need to remove the indentations that we have - namely, in the inner corners of our cross. How to do this? For example, cutting off some protruding triangles from the outer corners of the cross...