## FROM A HERRINGBONE. QUADRAT

Cutting problems are more complicated than puzzles. One cannot do without logical and spatial thinking. One also needs a sharp eye to see two or more images in one picture. For example, if you cut a **square** into two parts diagonally, you can make an isosceles triangle. This is, of course, one of the easiest problems. You can cut a square into three parts to make a rectangle. However, this task is not difficult. Let’s try a more difficult task: to reshape a Christmas tree by cutting it into five pieces so that they can be made into a square.

The problem is solved, but the question remains: Is it possible to make this herringbone square by cutting it into fewer pieces?? Finding the answer is almost always difficult. But if a shorter solution can still be found, it’s hard to resist surprise and delight: “This can’t be!!”

For example, the same Christmas tree can be “squared” by cutting it into four parts. This “squaring” was invented by Ekaterina Markina, a ninth-grader, at my lesson at the regional summer camp for gifted children. Perhaps her number of cuts is minimal.

Developed many ways and techniques of cutting. For example, you can reshape a Christmas tree into a square using two overlapping parquetries. Let’s specify that a parquet is a plane composed of several figures without gaps and overlaps. Try to see a parquet made of identical herringbones, and a parquet made of squares equal to the herringbone. The borders of the squares are mowing fishing line, along which you need to cut the Christmas tree into four pieces. And here you can see how to make a square from these parts. It’s really beautiful?

In practice, when solving cutting problems with this method, it is convenient to draw one of the parquet on paper, and the other on transparent foil or tracing paper. Laying the tape on the paper, select such an arrangement of one parcel relative to the other, which results in the smallest number of pieces. Those who know how to use a computer can impose parquetry in any graphics editor.

And now for the task. In Fig. 5 There are four Christmas trees. All of them are symmetrical. Each Christmas tree, having redrawn it before, must be cut into four parts so that they form a **square**.

Hint. Among these herringbones there are some that are easy to cut. for warming up, but there are some that are called “tough nuts”. Calculate the area of the herringbone, taking as a unit, for example, a notebook cell, and then find the side of the square. And here’s how to cut. think about it.

- A square has 16 cells. Divide the
**square**into two equal parts so that the cutting line runs along the sides of the cells. (The way of cutting a square in two parts will be considered different if the parts of the square obtained by one method of cutting are not equal to the parts obtained by the other method.) How many solutions do the problem have in total?? - A 3×4 rectangle has 12 cells. Find five ways of cutting the rectangle into two equal parts such that the line of cutting goes along the sides of cells (ways of cutting are considered distinct if the parts obtained by one way of cutting are not equal to the parts obtained by another way).
- A 3X5 rectangle contains 15 cells and the central cell is removed. Find five ways to cut the rest of the figure into two equal parts so that the fishing line runs along the sides of cells.
- A 6×6 square is dissected into 36 identical squares. Find five ways of cutting a
**square**into two equal parts so that the line of the cut goes along the sides of the squares. Note: The problem has more than 200 solutions. - Divide the 4×4
**square**into four equal parts so that the cutting line runs along the sides of the cells. How many different ways of cutting can you find? - Divide the figure (Fig.5) into three equal parts so that the fishing line cuts along the sides of the squares.

Divide this figure (Fig.6) into four equal parts so that the cutting line runs along the sides of the squares.

Divide the figure (Fig.7) into four equal parts so that the cut lines run along the sides of the squares. Find as many solutions as possible.

Divide the 5×5 **square** with the center cell cut out into four equal parts.

Cut the shapes shown in Fig.8 into two equal parts along the grid lines, and each part should have a circle in it.

The figures shown in Fig.9, you must cut along the grid lines into four equal parts so that each part has a circle. How to do this?

Cut the figure shown in fig.10, along the grid lines into four equal parts and make a square of them so that the circles and asterisks are located symmetrically about all the symmetry axes of the

square.

Cut the given square (Fig.11) along the sides of the squares so that all the pieces are the same size and shape and so that each contains one circle and an asterisk.

Cut up the 6×6 square of checkered paper in Fig.12, into four equal parts so that each of them contains three shaded cells.

## Cut the photo into equal parts online

The main thing is to specify the picture on your computer or phone, if necessary, specify how many parts should be in width and height, click OK, wait a couple of seconds, download the result. The other settings are already preset by default. There is also the usual photo cropping, where you can specify how many % or pixels you want to crop on each side.

Example of a photo before and after the cut into two equal parts vertically, the settings are set by default:

With this online service, you can cut a picture into two, three, four, five, or even 900 equal or square pieces, and automatically cut a photo for Instagram by specifying just the right cropping format, such as 3×2 for a horizontal photo, 3×3 for a **square** photo, or 3×4 for a vertical. If you need to process a huge picture of more than 100 megapixels, cut it into more parts or need a different numbering of sliced.If you need a jpg file, send it to me. It will be done for free in 24 hours.

The original image does not change. You will be given several pictures cut into equal parts.

## Summary of the lesson “Dividing a **square** into four equal parts” in the older group

Goal: To form the concept that a square can be divided into four equal parts.

*Educational:*. developing the ability to name the parts obtained by division, to compare the whole and the parts, to understand that the whole object is larger than each of its parts, and the part is smaller than the whole. Strengthening the ability to name and compare the whole and the parts.

-development of logical and figurative thinking, spatial imagination, thinking abilities of children, the idea of how to make one shape into another.

*Teaching methods:* Verbal, visual, play and practical.

Techniques: visual demonstration, practical actions, activation of attention, speech, questions, motor activity.

Organizing moment

Guys, we have guests, let’s smile at them and say hello. Dear children! We are both very big, and next year we will be the oldest in kindergarten, and very soon you will go to school. To do well in school, you need to know a lot, be able to think, do smart tricks and solve problems. Let’s try to do the tasks I have prepared for you today. YOU help me?

Come on guys, let’s sit at our tables. Each of you has a square on your table. Maybe someone knows how a **square** can be divided into two parts? (You can divide into 4 squares across by adding its sides in half and in half again; into 4 triangles by adding a diagonal corner to a corner, in half and in half again).

Game exercise “Divide a square into parts.”

*Teacher:* Guys, we need to divide the **square** into two and four equal parts. Who will show how to divide a **square** into two parts in different ways.

*A child comes out.* (Shows how a square can be divided into two rectangles)

(And the other way to triangles is shown by the teacher.)

*Educator.* What kind of shapes you have? (Rectangles and triangles.)

*Teacher.* How to name each part?

*Child.* One second of a **square**, a triangle.

*Provider.* Right half is one of 2 equal parts of the whole. Both equal parts are called halves. Each part is called one second or half because divided into two equal parts.

*Educator.* Which is bigger: the whole square or part of it??

*Educator.* Which is smaller: one second of the square or the whole **square**?

*Facilitator.* How to get four equal parts?

*Child:.* We need to cut each half in half again.

Challenge the children to fold and cut each half in half.

*Educator.* How many parts did each of you get?? What can you call each part?? (children’s answers).

*Provider.* Correct, each of the parts is called one fourth, so we divided the whole into four parts, also this part is called a quarter or one fourth.

*Educator.* Which is bigger: the whole square or one quarter?

*Provider.* Which is smaller: one-fourth of a **square** or one-second of a square?

*Provider.* Which is bigger: half a square or one fourth of a square??

*Educator.* Which is smaller: one-fourth of a square or one-second of a square? (Showing the parts to be compared.).

Each child has a square and a pair of scissors on the table.

*Educator.* Guys, divide the squares into four equal parts in different ways.

*Teacher.* How to divide a square into four equal parts?

*Child.* I will fold a square in half, precisely joining the sides and corners of the square, I will iron the line fold and cut evenly along the mowing line with scissors, then once again I will fold each part in half and cut evenly along the mowing line with scissors.

The children divide the squares into four equal parts.

*Educator.* Guys, who can tell us how many parts we divided the square into??

*The child.* We divided the square in half into four equal parts in different ways.

Let’s have a little rest, come out from the tables. Look at what geometric shape our group looks like? Let’s try to divide it in half (horizontally and vertically). And into four parts.

5 “Modeling an Object”.

Now I suggest you take one piece of paper and one glue stick each and sit in your seats.

Let’s make fun toys out of our quarters. I have handed out cards on the table and ask you think about using our quarters and halves to make toys like this? Choose your favorite toy and start (first you need to lay out your chosen picture on a sheet of paper, make sure you get it right and only then glue it on).

Turn the piece of paper over. The back side of your sheet is not white, but dotted.

Let’s draw on the cages, and you tell me what you get.

Put a dot in the middle of the sheet at the intersection of the cells like this.

Right 2 cells, down 4 cells, left 2 cells. Up 4 cells, left 2 cells, down 4 cells, right 2 cells.

Summary of the lesson in the middle speech therapy group “We are different, but we are equal!” Prospectus of the lesson of the middle speech therapy group. Topic: “We are different, but we are equal!”Integration of Educational Areas: Social and Communicative.

How to cut an image into equal parts to decorate a group” How I cut a picture and what I got out of it I don’t know about you, but I am not an artist. Pencils and paints are not my strong suit. But by the nature of their.

Outline of an integrated lesson in the older group “Man. Body parts” Outline of an integrated lesson in the older group “Human. Body Parts”. Cognitive Development”, “Art and Aesthetic.

Outline of the IFL in mathematics in the older group “Division of the square into parts”. Program objectives: To consolidate the counting within 10 (forward and backward). To find the next and previous number. To consolidate the ability of children to make.

The outline of the open class for FEMP in the middle group “One, two, three, four, five. learning to count” “One, two, three, four, five we learn to count.”The goal of the open class: To promote the formation of children’s numeracy skills up to 5. The course of the lesson.

Summary of the lesson in the preparatory group “Body Parts. Making a retelling” Objectives: In the course of direct educational activities is necessary: 1. To consolidate the generalizing concept of “body parts”; 2. To consolidate.

Outline of the lesson “We study the parts of the face” in the middle group Outline of the frontal lesson in the middle group on the theme: “The parts of my face. Objective: to enrich and activate lexico-grammatical structure.

## Master class Origami Divide the square into equal parts Paper

Dividing the square into equal parts. it is always only a preparatory stage for folding. However, without certain skills, just he can be quite difficult, especially if the number of parts, is a simple number:3, 5, 7, as well as 9. Let’s talk a little bit more about this.

Mark the center of the upper side. For this we make a small tack.

Fold the corner of the square to the middle of the opposite side.

In this case the intersection point of the side opposite to this corner and the side adjacent to it will divide the side in the ratio 1:2. So, using only the folds we have found one third of the side of the square.

Arranging the **square**. The tack on the left side is 1/3 of it.

Using this tack we make a fold. In doing so, it should be parallel to the upper and lower sides.

Turn the sheet to the opposite side.

### How to divide three squares into 4 equal parts?

We fold the resulting rectangle in half.

Thus we obtain three parallel folds. They have divided the square into three equal parts.

We divide the square into five equal parts.

Let’s mark with a tack the middle of the side.

Let’s make a fold that goes through the bottom left corner of the **square** and our mark at the same time. The bottom right corner is located horizontally 2/5 of the right edge.

Divide the resulting segment in half. The width of the bent strip is 1/5.

Straighten the sheet. Now we need to divide the rest of the sheet into four equal parts.

Fold the left side to the intended vertical fold. So we divide this space in half.

Straighten the sheet. It remains for each of the wide strips to divide in half more.

Fold the left side to the side fold that we identified in the previous step.

It remains, divide the last sector. To do this, align the right side line with the leftmost vertical fold.

We straighten the sheet. Dividing the sheet into five equal parts is now complete.

In order to divide the sheet into seven equal parts, you must first divide it into five, as described above.

Make the fold, in which the bottom right corner coincides with the second mark on the right.

Straighten the sheet. The point on the right side that is formed by this bend. This is 3 / 7 of the top edge or 4 / 7 of the bottom.

Align the bottom right corner with the point obtained on the right side. Perform a fold that will be parallel to the upper and lower sides.

Fold the underside to the resulting horizontal fold. The width of this strip will be 1/7th of the side.

Make the above crease “mountain” and combine it with the above mark, which divided the side of the 3/7 and 4/7.

Align the top side with the fold obtained in the previous step.

Unfold the sheet. It remains to divide each of two top rectangles in half more.

Combine the upper side with the fold obtained in the previous step.

Align the top side with the lowest horizontal fold.

Unfold the sheet. Our horizontal square is divided into seven equal parts.

In order to divide the sheet into nine equal parts, you must first divide it into three, as described above.

Make a fold where the bottom right corner aligns with the first mark on the right.

The point obtained on the right side will divide it by 4/9(top) and 5/9(bottom). Further division into equal parts can be different. Below is one way to complete the division of the square into equal parts.

With the point obtained on the right side, make a fold parallel to the top and bottom edge. The difference by which the bottom will be wider than the top. and it’s 1/9.

Flip to the opposite side.

Fold back the top layer of paper. The fold should coincide with the edge of the bottom layer.

Turn it back to the opposite side. Unfold the sheet.

The resulting fold in the previous step is combined with the line, which was obtained by using a tack.

Align the top edge with the same line. We have something like a basic form of “door”. Now it remains to divide each of the four wide rectangles into two more.

Unfolding the Sheet. Our horizontal **square** is divided into nine equal parts.

## Olympiad, logical and fun math problems. Cutting Tasks

For math tutors and teachers of various extracurricular activities and circles we offer a selection of entertaining and developing geometric problems for cutting. The goal of using such problems in his classes is not only to interest students in interesting and effective combinations of squares and shapes, but also to form a sense of lines, angles, and shapes. The problem set is primarily geared toward children in grades 4-6, although its use with even older students is not ruled out. The exercises require high and steady concentration from students and are perfect for developing and training visual memory. Recommended for mathematics tutors who prepare students for entrance exams to mathematics schools and classes that place special demands on the level of independent thinking and creativity of a child. The level of problems corresponds to the level of the entrance Olympiads of the Lyceum, the Second Mathematical School, the Junior Faculty of Mechanics and Mathematics of Moscow State University, the Kurchatov School, etc.

Math Tutor’s Note: Some solutions to problems, which you can see by clicking on the appropriate pointer, show only one possible example of cutting. I quite admit that you may get some other correct combination do not be afraid of it. Check your mooch’s solution carefully and if it satisfies the condition, then feel free to take on the next problem.

1) Try to cut the figure shown in the picture into 3 equally shaped pieces:

Math Tutor Hint: The small shapes are very similar to the letter T View Math Tutor Solution

2) Now cut this figure into 4 equal in shape parts:

Math tutor hint: It is easy to guess that the small figures will consist of 3 squares, and there are not many figures of 3 squares. There are only two kinds: a corner and a rectangle 1×3. View the math tutor’s solution:

3) Cut the given figure into 5 equal-shaped pieces:

Math Tutor Hint: Find the number of cells that make up each of these shapes. These figures look like the letter G. View the math tutor’s solution

4) Now you need to cut a figure of ten cells into 4 unequal rectangles (or squares).

Math Tutor’s instructions: Highlight any rectangle and then try to write three more into the remaining squares. If it doesn’t work, change the first rectangle and try again. View Math Tutor Solutions

5) The problem is more complicated: you need to cut the figure into 4 different shaped pieces (not necessarily into rectangles).

Math tutor tip: First draw all kinds of differently shaped shapes separately (there will be more than four of them) and repeat the enumeration method as in the previous problem. View math tutor solution:

6) Cut this figure into 5 pieces of four differently shaped cells so that only one green cell is shaded in each of them.

Math Tutor Hint: Try to start cutting from the top edge of a given figure and you will immediately know how to proceed. View Math Tutor’s solution:

7) Similar to the previous problem. Find how many differently shaped shapes there are that have exactly four cells? Figures can be twisted, rotated, but cannot be lifted off the sostool (from its surface) on which it lies. That is, the two figures given would not be considered equal because they could not be obtained from each other by rotation.

Math Tutor Hint: Study the solution to the previous problem and try to imagine the different positions of these shapes as they rotate. It is not difficult to guess that the answer in our problem is a number 5 or more. (Actually even more than six.). There are 7 types of shapes described. View Math Tutor Solutions

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

Math tutor tip: The shapes of the little figures are not a square or rectangle, and they are not even a corner of four cells. So what kind of shapes should we try to cut? See Math Tutor Solutions

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

Math tutor tip: There are 16 squares in all, so a square is 4 squares×4. And you also need to fill the box in the middle somehow. How to do this? Maybe by some kind of shift? Then because the length of the rectangle is equal to an odd number of squares, the cut must be made not by a vertical cut, but by a broken slash. So that the top part is cut from one side of the middle cell and the bottom part from the other. View Math Tutor Solution

10) Cut a rectangle of size 4×9 into two pieces so that the result can be made into a square.

Math tutor tip: A rectangle has a total of 36 squares. So the square will turn out to be 6×6. Since the long side has nine squares, three of them need to be cut off. How this cut will go further? View the math tutor’s solution

11) The five-cell cross shown in the figure needs to be cut (the cells themselves can be cut) into pieces such that they can be made into a square.

Math Tutor Hint: 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 problem in which you are allowed to cut outside of the squares. However, it is still a good idea to leave them as a reference. For example, it is worth noting that we somehow need to remove the indentations that we have, namely, in the inner corners of our cross. How would this be done? For example, by cutting some protruding triangles from the outer corners of the cross. View Math Tutor Solution: Comment: Cut as shown in the picture and put the blue triangles into the empty areas shown by the purple triangles.

Alexander Kolpakov. Tutor in Mathematics Moscow, Strogino.

Great site! Thank you for the most interesting problems with answers on the entire Internet!

## Answers to page 100 71-380 GDZ for the textbook Mathematics 5 class Merzlyak, Polonsky, Yakir

Rectangle ABCD dissected into squares as shown in Figure 139. The side of the smallest of the squares is 4 cm. Find the lengths of the sides of rectangle ABCD.

The side of the smallest square is 4 cm, 4 3 = 12 (cm). the side of the largest squareAD = BC = 12 12 4 = 28 (cm) The sides of AD and BC consist of 4 middle squares28 : 4 = 7 (cm). side of the middle squareCD = AB = 7 4 3 = 19 (cm)Answer: 28 cm and 19 cm.

### Problem 72

Draw a rectangle whose adjacent sides are 3 cm and 6 cm. Divide it into three equal rectangles. Calculate the perimeter of each rectangle obtained. How many solutions to the problem does the problem have?

The problem has 2 solutions: 1) AK = KM = MD = BN = NP = PC = 6 : 3 = 2 (cm) P ABNK = P KNPM = P MPCD = 2 2 2 3 = 10 (cm)

2) AK = KM = MD = BN = NP = PC = 3 : 3 = 1 (cm) P ABNK = P KNPM = P MPCD = 2 1 2 6 = 14 (cm)

### Task 73

Is there any rectangle with a perimeter of 12 cm which can be divided into two equal squares?? If the answer is YES, draw a diagram and calculate the perimeter of each of the squares obtained.

Rectangle ABCD with sides 4 cm and 2 cm. P ABCD = 2 2 2 4 = 4 8 = 12 (cm) By dividing the rectangle in half we obtain a square with a side of 2 cm, the perimeter of which is equal: P ABEF = P FECD = 2 4 = 8 (cm) Answer: 8 cm. The perimeter of each of the squares.

### Task 74

How to divide a square into four equal parts so that two squares can be made of these parts?

Then from each pair of triangles we add a square.

### Assignment 75

How to cut an isosceles right triangle into four equal pieces so that the pieces make a **square**?

### Problem 76

How to cut a rectangle with sides of 8 cm and 4 cm into four parts so that it could be made into a square?

### Task 77

How do you cut a square into a triangle and a quadrilateral so that you can form a triangle??

Put a triangle on top of a quadrilateral and you get a big triangle.

### Problem 78

How to cut a **square** with a side of 6 cm into two pieces using a polyline of three links so that the pieces could be made into a rectangle?

### Problem 79

Draw straight line MK, ray PS and segment AB so that the ray intersects the segment AB and the straight line MK, and the straight line MK does not intersect the segment AB.

### Assignment 80

There are lemons, oranges, and tangerines in the store, 740 kg in total. If 55 kg of lemons, 36 kg of oranges, and 34 kg of tangerines were sold, the remaining masses of lemons, oranges, and tangerines would be. How many kilograms of each type of fruit are in the store??

You can’t divide by a fraction because there’s a misprint in the problem. In principle, you could solve this problem using fractions:

1) 55 36 34 = 125 (kg). would have sold a total of 2) 740. 126 = 615 (kg). the fruit would be left in the store 3) 615 : 3 = 205 (kg). The mass of each of the remaining kinds of fruit 4) 205 55 = 260 (kg). lemons in the store 5) 205 36 = 241 (kg). oranges are available in the store 6) 205 34 = 239 (kg). Answer: 260 kg of lemons, 241 kg of oranges, 239 kg of tangerines.

## Distance courses for teachers

### The material is suitable for the UMK

“Mathematics. Visual Geometry”, I. Sharygin.Ф., Ergangieva L.Н.

Tasks on cutting and folding shapes

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Moscow Institute for Professional Retraining and Professional Development of Educators