You can walk the path without lifting your pencil. Building figures with one stroke of a pencil
Without lifting your pencil
Purpose: to teach students to identify, depict and compose geometric shapes,
which can be drawn without taking the pencil off the paper;
formulate signs of drawing figures in one stroke;
involve students in various activities: observation, research,
the ability to draw conclusions.
During the classes.
I. Introductory speech of the teacher:
Many people put their signature in a continuous line, and for each person it is specific. Are there any among you? (Show a sample of your signature.)
It is known from history that Mohammed (Muhammad, the founder of the Muslim religion), instead of a signature, described in one stroke a sign consisting of two horns of the moon: I hope that at the end of our lesson you can do it too.
Give examples of geometric shapes and letters of our alphabet that can be drawn without lifting the pencil (circle, square, triangle; G, L, M, P, S). Draw a triangle. To solve such problems, there are signs by which you can check whether this figure can be built without lifting the pencil from the paper. If so, from what point should this drawing be started?
There is a section in mathematics that studies the properties of such shapes (find the answer by solving the crossword keyword)
1. Part of a straight line (segment).
2. A figure consisting of two identical squares (dominoes).
3. The sum of the lengths of all sides of a triangle (perimeter).
4. Device for measuring angles (protractor).
5. Corners 1 and 2 _______ (vertical).
6. The end of these words is a mathematical term of 5 letters.
LAS
FOR (..) (dot).
LINEN
7. Unit of measurement of angles (degree).
8. A segment connecting the vertex of the triangle with the midpoint of the opposite side (median).
9. Author of the textbook "Geometry grade 7-9" (Atanasyan).
Topology is a branch of mathematics that studies such properties of figures that do not change when the figures are deformed without breaking or gluing.
For example, in terms of topology, a circle, an ellipse, a square, and a triangle have the same properties and are the same figure, since one can be transformed into another. But the ring is not one of these: to turn it into a circle, gluing is necessary.
A planar graph is a set of points in a plane.
The top of the graph - points of the plane, interconnected
Edges are lines that connect vertices.
Let's agree to call the vertex at which an even number of lines converge the word "even", and the vertex at which an odd number of lines converge - "odd".
A (n), C (n), B (h), D (h)
Conclusion:
1. if there are no odd vertices in the figure, then it can be drawn without lifting the pencil.
2. If there are no more than two odd vertices, then you can draw a figure, and you need to start at one of the odd vertices and end at the other (if the figure has one odd vertex, then it has a second one).
.
There are two envelopes on the board, one open, the other closed.
Task: redraw the envelopes in a notebook and outline them in a different color, adhering to the rule - do not tear the pencil off the paper and do not pass it twice along any line.
A-B-E-C-D-B-C-A-D
If there are no more than two odd points, then you can draw a figure, and you need to start at one of the odd points and end at the other (if the figure has one odd point, then it has a second one).
The figure shows various shapes. Establish which figures can be drawn without lifting the pencil from the paper, and which cannot.
Today's children are difficult to captivate with something. They like to watch cartoons and play computer games. But smart parents are always able to interest their child. For example, they may suggest that he find a way to draw an envelope without lifting his hand. Read about some tricks of this task below.
Warm up
Before you start torturing your child with logical tasks, you need to do preparatory work with him. Why is she needed? So that the child does not cheat when he starts to puzzle over the question of how to draw an envelope without taking his hands off. After all, the most interesting thing in this problem is that the line must go from point to point continuously.
What tasks can be offered to the child as a warm-up? Of course, the first one should be eights. Drawing this figure relieves stress, clears the brain, and trains the hand. All in all, a useful exercise. After that, you can move on to drawing rounded shapes. It can be curls or any other squiggles, the main thing is that in the process of drawing the child does not tear off the pencil and depict everything in one smooth line.
How to draw a closed envelope
Many parents themselves have spent more than one hour before offering such a task to a child. You can try too. But we can immediately upset you - it is simply impossible to complete such a task without being a little cunning. Therefore, we will tell you a method that will help you and your child go a little beyond the usual logic in order to understand how to draw a closed envelope without taking your hands off.
We take a sheet of paper and bend its edge. We bend it back. Now our task is to draw the top edge of the closed envelope just on the fold line. To make it easier to understand, let's place dots at the ends of the rectangle. Let's number them, starting from the upper left corner. Here will be the number one and then clockwise. From the number 4 to 1 we draw a line, now we connect 1 to 2 and now we draw a diagonal to 4. From 4 to 3 we draw a straight line, and then again a diagonal to 1.
Now we pass to the most interesting. We bend the edge of our sheet and depict a zigzag, which forms, as it were, the cap of our envelope. It will pass from 1 to 2. It remains to connect 2 and 3 with a straight line - and the puzzle is solved. Fold back part of the sheet. The riddle of how to draw an envelope without taking your hands off can be offered not only to children, but also to friends or colleagues.
How to draw an open envelope
Those who carefully read the previous paragraph and created their drawing according to the description already understood how to answer the question posed above. After all, the solution to the riddle of how to draw an open envelope without taking your hands off will be similar to that written in the previous paragraph. Only here you do not have to bend and bend parts of the sheet. The entire image will be made in one line in the same way.
But if you do not want to repeat yourself, then we offer another way that will lead to the same result. How to draw an envelope without taking your hands off the second way? To begin with, we draw a rectangle again with dots and number it again, as in the previous paragraph. From the number 4 to 2 we draw a diagonal, from 2 to 3 - a straight line, and from 3 to 1 - again a diagonal. Next you need to draw a corner. From 1 to 2, draw a zigzag that marks the top of the envelope. From 2 we return to 1 with a straight line and complete our construction by alternately drawing straight lines from 1 to 4 and from 4 to 3.
Why are these tasks necessary?
These should be done not only by children, but also by adults. Thanks to them, the human brain strains and begins to work. If you accustom yourself to perform a similar task every day, after a month you will notice that in critical situations solutions are generated faster and less effort is spent on it. It is especially useful for schoolchildren to study logic puzzles. In this way, they train creativity and learn to approach standard questions in a non-standard way.
Instruction
It is assumed that the given figure consists of points connected by straight or curved segments. Therefore, at each such point a certain segment converges. Such figures are called graphs.
If an even number of segments converge at a point, then such a point itself is called an even vertex. If the number of segments is odd, then the vertex is called odd. For example, a square in which both are drawn has four odd vertices and one even vertex at the intersection point of the diagonals.
A line segment, by definition, has two , and therefore always connects two vertices. Therefore, summing all the incoming segments for all the vertices of the graph, only an even number is possible. Therefore, whatever the graph, there will always be an even number of odd vertices in it (including zero).
A graph in which there are no odd vertices at all can always be drawn without taking your hand off the paper. It doesn't matter where you start from.
If there are only two odd vertices, then such a graph is also unicursal. The path must necessarily begin at one of the odd vertices, and end at the other of them.
A figure in which there are four or more odd vertices is not unicursal, and it is not possible to draw it without repetition of lines. For example, the same square with drawn diagonals is not unicursal, since it has four odd vertices. But a square with one diagonal or an "envelope" - a square with diagonals and a "lid" - can be drawn with one line.
To solve the problem, you need to imagine that each line drawn disappears from the figure - you cannot go through it a second time. Therefore, when depicting a unicursal figure, care must be taken that the rest of the work does not fall apart into unrelated parts. If it happens, it will no longer be possible to bring the matter to the end.
Sources:
- How to draw a closed envelope without taking your hands off?
Square is an equilateral and rectangular quadrilateral. It is very easy to draw it. Start training first on a notebook in a cage. With the help of a simple pencil and an invisible square from, learn how to draw a square without taking your hand off the paper.
You will need
- - a simple pencil;
- - sheet in a cage;
- - A4 sheet;
- - ruler.
Instruction
You can try this: without using a ruler and dots. Draw a square in the middle of the sheet. At first, don't try to draw it with four perfect lines. Draw the sides of the square "through", pointing additional lines until the square turns out to be a square. Do not take your hand off the paper. Draw lines parallel to the edges of the paper. Do some of these training exercises. This one will teach you straight lines and square without tearing hands.
Sources:
- square pattern
In the painted urban or rural landscapes, various bridges. This special building may look elegant and weightless, or, on the contrary, it may give the impression of a strict and heavy structure.
You will need
- pencil, paper, paints
Instruction
Equivalent and equidistant figures
With equal figures, one should not confuse equal-sized and equally-composed figures - with all the closeness of these concepts.
Equal-sized figures are those that have an equal area if they are figures on a plane, or an equal volume if we are talking about three-dimensional bodies. The coincidence of all the elements that make up these figures is not mandatory. Equal figures will always be equal in size, but not all figures of equal size can be called equal.
The concept of equiconsistency is most often applied to polygons. It implies that polygons can be divided into the same number of respectively equal shapes. Equivalent polygons are always equal area.
Sources:
- What are equal figures
I. Statement of the problem situation.
Probably, everyone remembers from childhood that the following task was very popular: without lifting the pencil from the paper and without drawing the same line twice, draw an “open envelope”:
Try drawing an “open envelope”.
As you can see, some succeed and some don't. Why is this happening? How to draw correctly to succeed? And what is it for? To answer these questions, I will tell you one historical fact.
The city of Koenigsberg (after the World War it was called Kaliningrad) stands on the Pregol River. Once there were 7 bridges that connected the shores and two islands. The inhabitants of the city noticed that they could not make a walk along all seven bridges, having passed over each of them exactly once. So a puzzle arose: “Is it possible to pass all seven Königsberg bridges exactly once and return to the starting place?”.
Try it, maybe someone will succeed.
In 1735 this problem became known to Leonhard Euler. Euler found out that there is no such way, that is, he proved that this problem is unsolvable. Of course, Euler solved not only the Königsberg bridge problem, but a whole class of similar problems, for which he developed a solution method. You can see that the task is to draw a route on the map - a line, without lifting the pencil from the paper, go around all seven bridges and return to the starting point. Therefore, instead of a map of bridges, Euler began to consider a diagram of points and lines, discarding bridges, islands and coasts as non-mathematical concepts. Here's what he got:
A, B are islands, M, N are coasts, and seven curves are seven bridges.
Now the task is to go around the contour in the figure so that each curve is drawn exactly once.
In our time, such schemes of points and lines have come to be called graphs, points are called graph vertices, and lines are called graph edges. Several lines converge at each vertex of the graph. If the number of lines is even, then the vertex is called even; if the number of vertices is odd, then the vertex is called odd.
Let us prove the unsolvability of our problem.
As you can see, all vertices in our graph are odd. To begin with, we will prove that if the traversal of the graph does not start from an odd point, then it must necessarily end at this point
Consider, for example, a vertex with three lines. If we came along one line, we went out along the other, and returned again along the third. There is nowhere to go further (there are no more ribs). In our problem, we said that all points are odd, which means that after leaving one of them, we must end up at the other three odd points at once, which cannot be.
Before Euler, it had never occurred to anyone that the bridge puzzle and other loop-walking puzzles had anything to do with mathematics. Euler's analysis of such problems "is the first sprout of a new branch of mathematics, today known as topology."
Topology- This is a branch of mathematics that studies such properties of figures that do not change with deformations produced without gaps and gluing.
For example, from the point of view of topology, a circle, an ellipse, a square and a triangle have the same properties and are the same figure, since one can be deformed into another, but the ring does not apply to them, since in order to deform it into a circle, adhesive is required.
II. Signs of drawing a graph.
1. If there are no odd points in the graph, then it can be drawn in one stroke, without lifting the pencil from the paper, starting from any place.
2. If there are two odd vertices in the graph, then it can be drawn in one stroke without lifting the pencil from the paper, and you need to start drawing at one odd point and finish at another.
3. If there are more than two odd points in a graph, then it cannot be drawn with one stroke of a pencil.
Let's go back to our open envelope problem. Let's count the number of even and odd points: 2 odd and 3 even, which means that this figure can be drawn in one stroke, and you need to start at an odd point. Try it, now everyone succeeded?
Let's consolidate the knowledge gained. Decide which figures can be built and which cannot.
a) All points are even, so this figure can be built starting from any place, for example:
b) This figure has two odd points, so it can be built without lifting the pencil from the paper, starting from the odd point.
c) This figure has four odd points, so it cannot be constructed.
d) Here all points are even, so it can be constructed starting from any place.
Let's check how you learned new knowledge.
III. Independent work on cards with individual tasks.
Exercise: check if it is possible to walk across all the bridges by walking over each of them exactly once. And if possible, then draw the path.
IV. Lesson results.
