# How to cut a square in 4

## FROM THE FIRST FIGURE. THE SQUARE

Cutting tasks are more complicated than puzzles. You have to use good logical and spatial reasoning as well as a sharp eye to see two or more pictures in one picture. For example, if we cut a square in two diagonally, we can make an isosceles triangle. It’s certainly one of the easiest things to do. You can cut the square into three parts so that they make a rectangle. However, this is not a difficult task either. Let’s try to tackle the more difficult task of reshaping 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: can this Christmas tree be made into a square by cutting it into fewer pieces?? Finding the answer is almost always difficult. But if a shorter solution is still found, it is difficult to refrain from surprise and delight: “This can not be!”

For example, the same Christmas tree can be “quadrupled” by cutting it into four pieces This kind of “quartering” was invented in my class at the regional summer camp for gifted children by Ekaterina Markina, a ninth-grader. Probably the minimum number of cuts.

Developed many ways and techniques of cutting. For example, a Christmas tree can be reshaped into a square using two parquet, overlapping each other Clarify that the 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 herringbones. The boundaries of the squares are mowing line, on which the herringbone should be cut into four parts. And here you can see how to make a square out of these parts. It’s really beautiful?

In practice, when solving cutting problems in this way, it is convenient to draw one of the squares on paper, and the other on transparent foil or tracing paper. Putting the tape on the paper, select such an arrangement of one parquet in relation to the other, which results in the smallest number of pieces. Those who own a computer can apply the parquet in any graphic editor.

And now the task. In Fig. 5 depicts four herringbones. They’re all symmetrical. Each Christmas tree must be cut into four pieces so that they form a square.

Hint. Some of these Christmas trees are easy to cut apart for warming up, but some are “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. How to cut, though, think.

## How to cut a square into 4 equal parts

How should a given rectangle be cut into two pieces so that they can be made into: 1) a triangle, 2) a parallelogram (other than a rectangle), 3) a trapezoid?

Given a rectangle whose base is twice the height. 1) How do you cut a given rectangle into two pieces so that they can be made into an isosceles triangle? 2) How to cut this rectangle into three pieces that would make a square?

### How to drill square hole with plywood/Self interior

How can an equilateral triangle be dissected into: 1) two equal triangles; 2) three equal triangles; 3) four equal triangles; 4) six equal triangles; 5) eight; 6) twelve?

Given two equal squares. How to cut each of them into two parts so that the resulting parts can be made into a square?

How to split this rectangle into two straight cuts into two equal pentagons and two equal rectangular triangles?

Given two unequal squares. How do you cut them into pieces so that the third square can be made of them?? How is the side length of the third square expressed in terms of the side lengths of the two given?

A rectangular bar of chocolate consists of mn unit square slices. How many breaks must be made (one piece breaks at a time) to break this tile into unit square slices?

How many cuts must be made by planes so that a cube with an edge of 3 dm can be made into cubes with an edge of 1 dm?

Given a right triangle. How should it be dissected into two such pieces that from them (without flipping them over) it is possible to form a triangle symmetric to this triangle with respect to one of its cathetuses?

Given triangle ABC. How to cut it into pieces so that from them (without flipping back) could be folded a triangle symmetrical to the given base AC?

Cut a square into pieces as shown in Figure 49, mix them up, and then add: 1) the same square; 2) a right-angled isosceles triangle; 3) a rectangle other than a square; 4) a parallelogram other than a rectangle; 5) a trapezoid.

A painted cube with an edge of 10 cm is sawn into cubes with a diamond of 1 cm. How many cubes would be made: 1) with one edge painted; 2) with two edges; 3) with three edges; 4) with no edges painted at all?

How to cut a rectangle with sides of 16 and 9 cm into two pieces so that it can be made into a square? (The cut can be in the form of a broken scythe line.)

Copy each of the figures in Figure 50 and cut it into 4 equal parts.

1) How to cut a given right-angled triangle into sharp-angled triangles? 2) How to dissect a given arbitrary triangle into sharp-angled triangles?

There are 10 points marked inside the convex stoop of which no three are on the same straight line. A polygon is dissected into triangles such that their vertices are all vertices of the given triangle and all ten given points. How many triangles will turn out?

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

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

Educational:. forming the ability to name the parts obtained by dividing, comparing the whole and the parts, understanding that the whole object is larger than each of its parts, and the part is smaller than the whole. Reinforcement of the ability to name parts, to compare the whole and the parts.

-to develop 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.

Organizational moment

Guys, we have visitors, let’s smile at them and say hello. Dear children! You and I are big, but next year we will be the oldest in kindergarten and soon you will go to school. In order to do well at school, you need to know a lot, be able to think, do the tasks of ingenuity, solve problems. Let’s try to do the exercises I have prepared for you today. YOU will help me?

Guys, let’s sit down at our tables. You each have squares on your desks. Maybe someone knows how a square can be divided into two parts and four parts? (It is possible to 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”

Educator: Guys, we need to divide the square into two and four equal parts. Who will show how to divide the 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 did you get? (Rectangles and triangles.)

Educator. What can you call each part?

The Child. One half of a square, a triangle.

Provider. Correct 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 half of the square or the whole square?

Provider. How to get four equal pieces?

Child. You have to cut each half in half again.

Challenged children fold and cut each half in half.

Provider. How many parts do each of you have?? What do you call each part? (children’s answers).

Educator. That’s right each part is called a 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?

Educator. Which is smaller: one fourth of a square or one second 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 being compared.).

Each child has a square and a pair of scissors on his or her desk.

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

### Circular Saw. Speed Square. Cut a Two By Four Square

Educator. How to divide a square into four equal parts?

Child. I will fold the square in half, exactly joining the sides and corners of the square, iron the folding line and cut evenly along the scythe line with scissors, then once again I will fold each part in half and cut evenly along the scythe line with scissors.

Children divide the squares into four equal parts.

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

Child. We divided the square in half into four equal parts in different ways.

Let’s take a little break and leave 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 pieces.

5 “Modeling the Object.”.

Now I suggest that 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 gave you on the table cards, think about whether our halves and quarters to make such toys? Choose your favorite toy and proceed (first you need to lay out your chosen picture on a sheet of paper, make sure you have it right and only then glue).

Turn over the leaf. The back of your worksheet is not white but squares.

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

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

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

Outline of the lesson of the middle speech therapy group “We are different, but we are equal!” The outline of the lesson of the middle speech therapy group. Subject: “We are different, but we are equal!” Integration of educational areas: social and communicative.

How to Cut an Image into Equal Sections for Group Design” tutorial How I cut pictures and what came out of it I don’t know about you, but I’m not an artist. Pencils and paints are not my strong suit. But by the nature of its.

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

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

The outline of the open class on FEMP in the middle group “One, two, three, four, five. learning to count” “One, two, three, four, five we learn to count.” The purpose of the open class: To promote the formation of children’s numeracy skills up to 5. Summary of the concept “parts of the body”; 2.

Summary of the lesson in the preparatory group “Parts of the body. Making a retelling” Objectives: During the direct educational activity it is necessary: 1. Strengthen the generalizing concept of “body parts”; 2. To consolidate.

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

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

Rectangle ABCD is cut into squares as shown in picture 139. The side of the smallest square 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). side of the largest squareAD = BC = 12 12 4 = 28 (cm) 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 of these rectangles. How many solutions to the problem do we have?

This problem has two 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)

### Problem 73

Is there any rectangle with a perimeter of 12 cm that can be divided into two equal squares?? If so, make a drawing and calculate the perimeter of each of the resulting squares.

Rectangle ABCD with sides 4 cm and 2 cm. P ABCD = 2 2 2 4 = 4 8 = 12 (cm) Dividing a rectangle in half will produce 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.

### Assignment 74

How do you divide a square into four equal parts so that you can make two squares??

Then we form a square from each pair of triangles.

How do you cut an isosceles right triangle into four equal parts so that you can make a square out of them?

### Assignment 76

How to cut a rectangle with sides 8 cm and 4 cm in four parts, so that they can be folded into a square?

### Problem 77

How do you cut a square into a triangle and a quadrilateral to make a triangle?

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

### Problem 78

How do you cut a square with a side of 6 cm into two pieces using a polyline consisting of three links so that the resulting pieces can be made into a rectangle?

### Assignment 79

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

### Assignment 80

The store has lemons, oranges, and tangerines, for a total of 740 kilograms. If we had sold 55 kg of lemons, 36 kg of oranges, and 34 kg of tangerines, the remaining masses of lemons, oranges, and tangerines would be equal. How many kilograms of each kind of fruit are in the store??

We can’t divide by two because there is a misprint in the problem. In principle, it is possible to solve this problem using fractions:

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

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