Presentation for the lesson of visual geometry in grade 5. Focused on a textbook for educational institutions "Visual Geometry", grades 5-6 / I.F. Shaprygin, L.N. Erganzhieva - Publisher: Drofa, 2015
Key concept: equality of figures. Subject results: depict equal figures and justify their equality; construct given figures from flat geometric figures; create and manipulate the image: dismember, rotate, combine, overlay. Meta-subject results: the development of imaginative thinking, design abilities, the ability to anticipate the result, the formation of communication skills.
Personal results: development of cognitive activity; instilling a taste for mental work. Intra-subject and inter-subject communications: planimetry (equality of figures, symmetry, area, equal size and equal composition), geometric combinatorics, drawing, technology.
This lesson is the first of two on this topic.
This lesson deals with cutting shapes. The goal of the solver is to cut the indicated figure into two or more equal parts. Often, for simplicity, this figure is divided into cells. In these problems, the concept of equality of figures is implicitly introduced (figures that coincide when superimposed are called equal). This definition is also used to check the equality of the resulting figures.
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Problems for cutting and folding figures. Lesson 1"
Cutting tasks
and folding figures
Purpose: to consolidate the ability to solve cutting problems.
visual geometry
5th grade
This proverb warns you against haste in solving problems.
A given figure, which is divided into equal cells for ease, must be cut into two or more parts.
If these parts can be superimposed one on another so that they coincide (while it is allowed to turn the figures over), then the problem is solved correctly.
Problem solving
Local land dealer
snatched off a piece of unusual land
forms (he expected to profitably sell it in parts).
But each of the eight found
im buyers, wanted to have
The plot is not worse than the neighbor's.
Where the merchant should install
dividing fences,
to get 8
the same areas?
Answer
Problem solving
A square consists of 16 identical cells,
4 of them are painted over. Cut the square into
4 equal parts so that in each of them
There was only one shaded cell.
A cell can occupy any place in each part.
Answer (4)
Problem solving
Cut the rectangle into 4 equal parts,
(use as many ways as you can).
1 way
The presentation offers only 4 ways to solve this problem. Perhaps students will suggest other ways - these should also be considered in the lesson.
2 way
3 way
Make shapes out of them. How many did they get?
The resulting
figures are called
TRIMINO .
Take four identical squares. Make shapes out of them.
- How many did they get?
Got five
TETRAMINO figures.
Make up of five squares
all possible figures.
How many did they get?
Total exist 12 pentomino elements
For the attention of tutors in mathematics and teachers of various electives and circles, a selection of entertaining and developing geometric cutting tasks is offered. The purpose of using such tasks by a tutor in his classes is not only to interest the student in interesting and effective combinations of cells and shapes, but also to form in him a sense of lines, angles and shapes. The set of tasks is mainly aimed 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 steady concentration of attention and are great for developing and training visual memory. Recommended for math tutors preparing students for entrance exams to math schools and classes that place special demands on the level of independent thinking and creativity of the child. The level of tasks corresponds to the level of introductory Olympiads in the lyceum "second school" (second mathematical school), small Mekhmat of Moscow State University, Kurchatov school, etc.
Math tutor's note:
In some problem solutions, 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 get some other correct combination - do not be afraid of this. Check carefully the solution of your mouse and if it satisfies the condition, then feel free to take on the next task.
1) Try to cut the figure shown in the figure into 3 equal parts:
: Small figures are very similar to the letter T
2) Now cut this figure into 4 equal parts:
Math tutor hint: It is easy to guess that small figures will consist of 3 cells, and there are not so many figures of three cells. There are only two types of them: a corner and a 1 × 3 rectangle.
3) Cut this figure into 5 equal parts:
Find the number of cells that each such figure consists of. These figurines look like the letter G.
4) And now you need to cut the figure of ten cells into 4 unequal rectangle (or square) to each other.
Indication of a tutor in mathematics: Select a rectangle, and then try to enter three more in the remaining cells. If it doesn't work, then 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 into rectangles).
Math tutor hint: first draw separately all kinds of shapes of different shapes (there will be more than four of them) and repeat the method of enumeration of options as in the previous task.
:
6) Cut this figure into 5 figures of four cells of different shapes so that only one green cell is filled in each of them.
Math Tutor Tip: Try to start cutting from the top edge of this shape and you will immediately understand how to proceed.
:
7) Based on the previous problem. Find how many figures of various shapes are there, consisting of exactly four cells? The figures can be twisted, rotated, but it is impossible to raise the sostole (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 of the previous problem and try to imagine the different positions of these figures when turning. It is easy to guess that the answer in our problem will be the number 5 or more. (In fact, even more than six). There are 7 types of described figures in total.
8) Cut a square of 16 cells into 4 equal parts so that each of the four parts has exactly one green cell.
Math tutor hint: The appearance of small figures is not a square or a rectangle, and not even a corner of four cells. So what shapes should we try to cut into?
9) Cut the depicted figure into two parts so that a square can be folded from the resulting parts.
Math tutor hint: In total, there are 16 cells in the figure, which means that the square will be 4 × 4 in size. And somehow you need to fill the window in the middle. How to do it? Maybe some kind of shift? Then, since the length of the rectangle is equal to an odd number of cells, 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 from the middle cells, and the lower part on the other.
10) Cut a 4×9 rectangle into two parts so that as a result you can add a square from them.
Math tutor hint: There are 36 cells in the rectangle. Therefore, the square will be 6 × 6 in size. Since the long side consists of nine cells, three of them need to be cut off. How will this cut go?
11) The cross of five cells shown in the figure needs to be cut (you can cut the cells themselves) into such parts from which a square could be folded.
Math tutor hint: It is clear that no matter how we cut along the lines of the cells, we won’t get a square, since there are only 5 cells. This is the only task in which it is allowed to cut not in cells. However, it would still be good to leave them as a guideline. for example, it is worth noting that we somehow need to remove the recesses that we have - namely, in the inner corners of our cross. How would you do it? For example, cutting off some protruding triangles from the outer corners of the cross...
Breakdown on checkered paper.
This is actually a simplified version of the Katamino game, requiring only checkered paper and a pencil. Such tasks are often found in textbooks and assignments for Olympiads for younger students. It is necessary to divide the figure drawn by cells into a given number of identical parts.
These tasks are suitable for a very wide age range, starting at three or four years old. But do not abuse them - they eventually get bored. Most likely, it is worth stopping at the complexity of 4-5 parts of 4-5 cells each.
Level 1
Rice. 1: Divide along the grid lines (by cells) into 2 equal parts.
Rice. 2: Divide along the grid lines into 3 equal parts.
Your children may need more simple tasks. They are very easy to compose: you just need to go "from the answer", i.e. take checkered paper, choose the shape of a figure ("part") from several cells and draw several such figures side by side, "blinding" them together. (It would be nice not to confuse the figures with their mirror images.) It doesn't matter if it turns out that the puzzle has two or more solutions - that means you need to find at least one (or all). Redraw the contour of the "monster" you have obtained on a blank sheet of checkered paper - the task is ready.
Level 2
Rice. 3: Divide the cells into 2 equal parts so that each of them has one
Red Square. (An additional condition - a red square - prohibits "extra"
solutions.)
Rice. 4: Divide along the grid lines into 3 equal parts.
Rice. 5: Divide along the grid lines into 4 equal parts.
Level 3
Rice. 6: Divide into 4 equal parts.
Introductory speech of the teacher:
A little historical background: Many scientists have been fond of cutting problems since ancient times. Solutions to many simple cutting problems were found by the ancient Greeks, the Chinese, but the first systematic treatise on this topic belongs to the pen of Abul-Vef. Geometers began to seriously tackle the problem of cutting figures into the smallest number of pieces 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.
Today, puzzle lovers are fond of solving cutting problems first because there is no universal method for solving such problems, and everyone who undertakes to solve them can fully show their ingenuity, intuition and ability to think creatively. (In the lesson, we will indicate only one of the possible examples of cutting. It is possible that students may get some other correct combination - do not be afraid of this).
This lesson is supposed to be carried out 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, scissors. Only straight cuts are allowed with scissors. Having cut some figure into parts, it is necessary to compose another figure from the same parts.
Cutting tasks:
1). Try to cut the figure shown in the figure into 3 equal parts:
Hint: The small shapes are very similar to the letter T.
2). Now cut this figure into 4 equal parts:
Hint: It is easy to guess that small figures will consist of 3 cells, and there are not so many figures of three cells. There are only two types: corner and rectangle.
3). Divide the figure into two identical parts, and fold the chessboard from the resulting parts.
Hint: Offer to start the task from the second part, how to get a chessboard. Recall what shape a chessboard (square) has. Count the number of cells in length, width. (Remind that there should be 8 cells).
4). Try three strokes of the knife to cut the cheese into eight equal pieces.
Hint: try cutting the cheese lengthwise.
Tasks for independent solution:
1). Cut out a paper square and do the following:
· cut into such 4 parts, from which you can make two equal smaller squares.
cut into five parts - four isosceles triangles and one square - and fold them so that you get three squares.