Lesson topic: Depicting relationships using a graph. Comparison

26.04.2024 | Chemistry

A person can tell not only about the properties of an object, but also about relationships, in which this object is located with other objects.
For example:
“Ivan is the son of Andrei”;
“Everest is higher than Elbrus”;
“Winnie the Pooh is friends with Piglet”;
"21 is a multiple of 3";
“Kostroma is as old a city as Moscow”;
“The word processor is part of the computer software.”

In each of the above sentences, the name of the relationship is highlighted, which denotes the nature of the connection between two objects.

Relationship can exist not only between two objects, but also between an object and many objects, for example:
“The floppy disk is a storage medium”;
"Kamchatka is a peninsula (is a peninsula)."

Each of these proposals describes attitude "is an element of a set".

A relation can connect two sets of objects, For example:
“Wheels are part of cars”;
“Butterflies are insects (are a type of insect).”

Several objects can be connected in pairs by the same relationship. The corresponding verbal description can be very long and then difficult to understand.

Let's talk about settlements A, B, C, D, D and E it is known that some of them are connected by railway: settlement A connected by railway to populated areas B, D and E, locality E- with populated areas B, D and D.

For greater clarity, the existing connections (“connected by railroad”) can be depicted with lines on the relationship diagram. Objects in a relationship diagram can be depicted as circles, ovals, dots, rectangles, etc. (Fig. 1.2).

Some relationship names change when object names are swapped, For example: “above” - “below”, “he is the father” - “he is the son.” In this case, the direction of the relationship is indicated by an arrow on the relationship diagram.

So, in Fig. 1.3, each arrow is directed from the father to his son and therefore reflects the relationship “is the father” and not “is the son.” For example: “Andrey is Ivan’s father.”

You don’t have to use arrows if you can formulate and follow the rule for the relative arrangement of objects on the diagram. For example, if in Fig. 1.3 Always place the names of children below the name of their father, then you can do without the arrows.

Such relationship, How “is his son”, “connected by railway”, “buys”, “treats”, etc. , can only connect objects of certain types. And in relationships“part of” and “is a variety” any objects can be located.

Briefly about the main thing

A message about an object can contain not only the properties of this object, but also the relationships that connect it with other objects. The name of the relationship denotes the nature of the relationship. Relationships can connect not only two objects, but also an object with a set of objects or two sets.

Any relationship between objects can be visually described using a relationship diagram . Objects in a relationship diagram can be depicted as circles, ovals, dots, rectangles, etc. Relationships between objects can be depicted as lines or arrows.

Questions and tasks

1. Name the relationship in each sentence given. What name can be given to the relationship if the names of the objects in the sentence are swapped? In which pairs the name of the relationship will not change?
a) Kolobok sings a song to the Fox.
b) The Little Humpbacked Horse helps Ivan.
c) There is Manezhnaya Square in Moscow.
d) Pilyulkin treats Siropchik.
e) The Scarecrow travels with Ell and...

2. For each pair of objects, indicate the corresponding relationship.

3. What connection is reflected in each relationship diagram in Fig. 1.4-1.8? Choose the correct answer from the following options:
"is a variety";
“included”;
“is a condition (cause)”;
"preceded by".

Types of objects and their classification

From two sets, related by the relation “is a variety” , one is a subset of the other. For example, the set of parrots is a subset of the set of birds, the set of natural numbers is a subset of the set of integers.

We will call the schema of the relation “is a variety” the schema of varieties(Fig. 1.9). Such diagrams are used in textbooks, catalogs and encyclopedias to describe a wide variety of objects, such as plants, animals, complex sentences, vehicles, etc.

In a variety diagram, the name of a subset is always located below the name of the set that contains it.

Objects of a subset necessarily have all the characteristics of objects of a set(inherit the characteristics of many) and in addition to them have their own additional characteristic (or several characteristics). This additional feature can be a property or an action. For example, any domestic animal needs to be fed, dogs, in addition, bark and bite, and sled dogs, in addition, also run in a harness.

It's important to understand, What objects themselves are not divided into any sets or subsets. For example, watermelon is completely “indifferent”; it is classified as a member of the pumpkin plant family, a subset of striped or spherical objects. Subsets of objects are identified and designated by a person, because it is more convenient for him to assimilate and transmit information. The fact is that a person can simultaneously concentrate his attention on only 5-9 objects. To simplify working with many objects, it is divided into several parts; each of these parts is again divided into parts; those, in turn, again, etc. The division of a large set into subsets does not occur spontaneously, but according to some characteristics of its objects.

A subset of objects that have common characteristics is called a class. Set divisionobjects into classes is called classification. The characteristics by which one class differs from another are called the basis of classification.

Classification is called natural if the essential characteristics of objects are taken as its basis. An example of natural classification is the classification of living things proposed by Carl Linnaeus (1735). Currently, scientists divide the variety of all living things into five main kingdoms: plants, fungi, animals, protozoa and prokaryotes. Each kingdom is divided into levels - systematic units. The highest level is called a type. Each phylum is divided into classes, classes into orders, orders into families, families into genera, and genera into species.

The classification is called artificial, if insignificant features of objects are taken as its basis. TO artificial classifications include auxiliary classifications (alphabetical subject indexes, name catalogs in libraries). An example of an artificial classification is the division of many stars in the sky into constellations, carried out according to characteristics that had nothing to do with the stars themselves.

We can propose the following classification of objects with which the user interacts in the Windows operating system (Fig. 1.10).

Briefly about the main thing

Varieties scheme is a diagram of “is of a kind” relationships between sets and subsets of objects.

Objects of a subset have additional characteristics, in addition to those that the objects of the set that includes this subset have.

A subset of objects that have common characteristics is called a class. Dividing a set of objects into classes is called classification. The characteristics by which one class differs from another are called the basis of classification.

Questions and tasks

1. For each of the indicated subsets, name the set with which it is related by the relation “is a variety” (name the general name that answers the question “What is it?”):
a) pronoun;
b) comma;
c) joystick;
d) parallelogram;
e) town hall;
f) fable;
g) capillary.

2. Find in the list six pairs of sets between which the “is a variety” relationship exists. Determine the name of the subset in each such pair. Name at least one additional property for it:
book;
petrol;
doctor;
milk;
builder;
textbook;
liquid;
directory;
Human.

3. Select from the list the names of nine sets related by the “is a kind” relationship. Make a diagram of the varieties:
Apple tree;
conifer tree;
pine;
fir;
tree;
deciduous tree;
apple;
trunk;
fruit tree;
birch;
oak;
larch;
root;
acorn.

4. Using the proposed classification of parallelograms, describe the properties of a square, which inherits them from two ancestors at once - a rectangle and a rhombus. What additional properties does a square have:
a) in relation to a rectangle;
b) in relation to a rhombus?

5. Each paragraph lists objects grouped by class. For example: table, computer, bow/cow, pen, pan/village, banner, feather - these are nouns classified by gender. Determine the basis of classifications:
a) spruce, pine, cedar, fir/birch, aspen, linden, poplar;
b) potatoes, onions, cucumbers, tomatoes/apples, oranges, pears, tangerines;
c) rye, silence, lie, lynx/wheat, silence, truth, cat;
d) shirt, jacket, dress, sundress/coat, fur coat, raincoat, windbreaker;
e) wolf, bear, fox, elk/cow, dog, cat, horse.

6. Offer your classification of computer objects “file” and “document”.

Practical work No. 2
“Working with file system objects”

1. Open the window My computer. Browse files and folders located on the disk WITH:.

2. Use the buttons Forward and Backward on the toolbar Regular buttons to move between previously viewed objects.

3. Select from menu Command View: Page Thumbnails, Tile, Icons, Table. Watch for changes in the display of folders and files. Find a button on the Common Buttons toolbar that allows you to quickly change the appearance of the contents of folders.

4. Using a button Folders display a panel on the left side of the window Folder Browser. Use it to once again view the files and folders located on the disk WITH:. Observe the changes occurring on the right side of the window.

5. Using a button Search Find your own folder - the folder where your work is stored. To do this in the window Assistant by search click on the link Files and folders. In the appropriate fields, specify the folder name and search area.

6. Open your own folder. It should contain subfolders Documents, Blanks_6, Blanks_7, Presentations and Drawings. View the contents of these folders.

7. The Worksheets_6 folder contains files that you used when completing computer practical work last year. Since you no longer need this folder, delete it (for example, using the context menu command).

8. The Documents, Presentations and Drawings folders contain your work from last year. I would like to save them.

Create an Archive folder in your own folder. To do this, move the mouse pointer to a blank area of the window of your own folder and right-click (call the context menu). Run the command [Create a folder].

Move the Documents, Presentations and Drawings folders one by one to the Archive folder. For this:
1) select the Documents folder and, while holding down the left mouse button, drag the Documents folder to the Archive folder;
2) open the context menu of Punks Presentations, execute the Cut command. Open the Archive folder and use the context menu to paste the Presentations folder into it;
3) cut the Pictures folder and paste it into the Archive folder using the menu bar commands.

9. Using the context menu, rename the Blanks_7 folder to Blanks.

10. Make sure your folder has a structure similar to the one below:

Standing one unit to the left and right of a given number. After this, the children will easily name the required numbers: for 7, these will be the numbers 6 and 8, for 11, these will be 10 and 12, etc. Exercise 24. Problem with missing data: it is unknown how many stamps were stuck on each envelope. For definiteness, we will assume that one stamp was affixed to each envelope. Solution: 12 – 6 = 6. Exercise 27. We remind you that for now the problem is solved without using subtraction. We reason like this: “The picture shows pairs of carrots and radishes.” 7 radishes left without pairs. This means that there are 7 more radishes than carrots, and 7 fewer carrots than radishes.” You can also reason like this: “To make all the pairs, 7 carrots were not enough. This means that there are 7 fewer carrots than radishes.” It is useful for students to provide such explanations themselves. He gave Yulia 7 postcards (12 without 5). Solution: 12 – 5 = 7. Answer: 7. Exercise 14 is used to develop graphic skills. This task is completed by students independently.< » и « > " to record the results of comparisons of numbers are not introduced in the first grade. Instead, colored arrows are used: red stands for the word “more”, and blue for “less”. You can compare not only two, but also more numbers. The result is drawings called graphs in mathematics. Next, ask the questions formulated in the text; After the children answer them, read the rule. They do not need to memorize this rule word for word. Do similar work with exercise 8 on p. 63 textbooks. Items can be compared by their price, that is, you can find out which one is more expensive or cheaper than the other. We compared numbers, found out which one was greater or less than the other, and expressed the results of the comparison in words. Sentences were obtained (in mathematics they are called statements). For example: “Yura is taller than Kolya”, 86 “An umbrella is cheaper than a raincoat”, “Three is less than six”, “Eight is more than zero”. Today you will learn how to briefly write down such statements. Let's agree to draw a red arrow instead of the words more, higher, older, longer, and instead of the words less, lower, younger, shorter - a blue one. Look at the chalkboard. It briefly contains several correct statements about numbers. The blue arrow replaces the word less, and the red arrow replaces more: p. With. k.k. 5 7 9 6 10 5 2 8 Let's read each of these statements. At the same time, let us remember that when reading a statement, we first name the number from which the arrow goes, then, moving along the arrow, we pronounce the word (“more” or “less”), and then we name the number to which the arrow goes. Let's try to read the first statement: which number we call the first (five), which word we pronounce (“less”), which number we call the second (seven). What happens? (Five is less than seven.) Now read the rest of the statements yourself.” Explain to students that in each of the tasks: “Read the statements,” three numbers are compared in pairs: 1 and 3, 3 and 8, 1 and 8, first in the “less than” ratio, then in the “more than” ratio. Pay their attention to the fact that the pictures differ in color and direction of the arrows. We read the statements: “One less than three”, “Three less than eight”, “One less than eight”; “Three is more than one,” “Eight is more than three,” “Eight is more than one.” Exercise 9. In this case, numbers are compared in pairs, and everywhere the arrows go from smaller numbers to larger ones. This means that all the arrows are blue. What is missing is an arrow going from 0 to 2 (0 is less than 2). ) Name the larger number (5), the smaller number (3). What action are we performing? (Subtraction.) From which number will we subtract which number? (Subtract 3 from 5.) How much will it be? (2.)” Solve the first few examples with detailed analysis. In the distant 90

I Organizing time

II Updating knowledge. Verbal counting.

· Count to 20 and back.

· Count from 11 to 19.

· Count from 16 to 7.

· What number is 2 units to the left of 15?

· What 2 numbers come after the number 18?

· Numbers got lost. Find these numbers and restore order.

· Name the numbers in this series: a) greater than 17; b) less than 7.

· How to determine on a scale a number greater or less than a given one?

· Which number is called first when counting: the larger or the smaller?

· Which number is greater: 5 or 6? Why?

· Which number is less than 32 or 23? Why?

Result: - I see that you remember how to compare numbers.

III Leading dialogue.

Can you compare objects by size? What words do you use for this?

What words do you use when comparing objects by height?

What if you compare objects by length?

When we compare something, we make sentences or statements. For example: “Seryozha is taller than Kolya”, “A textbook is more expensive than a notebook.”

Try to make a statement with the given word “cheaper”.

What about the word “younger”?

What have you compiled now?

What are these sentences called in the language of mathematics?

IV Lesson topic message.

Today you will learn how to represent statements graphically.

V Problematic question.

What do you think can replace the words “less” and “more”?

VI Discovery of new knowledge

Let's check your assumptions. (Remove and turn over the full houses: larger, higher, longer, heavier - on the back the graph is red. Similarly with the full houses smaller, lower, shorter, lighter - the graph is blue).

Conclusion: - So, to denote the words bigger, higher, longer, heavier, we use a red graph, and to denote the words smaller, lower, shorter, lighter - blue graph.

Working from the textbook pp. 90-91 No. 1. Let's make the first sentence.

Remember: first we name the object from which the arrow comes, then we say the word that is written above the arrow and name the object to which the arrow approaches.

Who will read the first statement?

Reading output

VII Work according to the textbook. Training exercises.

P.91, no. 2 - 3.

IX Work in a notebook. Exercises in graphical representation of relationships.

P. 60-61, No. 1 – 3

Result: - How to graphically represent the word “more”? What about the word “less”?

X Repetition and consolidation of what has been learned.

Work in notebook No. 3 No. 6.

What is the most convenient way to add the numbers 3 and 9?

What rule do you know?

Check the examples that need to be solved based on this rule.

Work according to the textbook.

Page 93 No. 9 (working with geometric material).

Tell us how we measure segments? Which segment is the longest? Which is the shortest?

Compare the lengths of the green and blue segments. Which arrow will we use to denote this relationship?

Page 93 No. 10 (working with a table)

Find answers to the questions using the data in the table.

Page 94 No. 18

How will we reason when solving this problem?

The ratio 11 greater than 10 is represented graphically.

XI Lesson summary.

What new did you learn in the lesson?

What tasks did you like?

Math lesson

Subject: Solving problems involving increasing and decreasing numbers by several times(lesson of generalization and systematization of knowledge)

Goals: creating conditions for developing the ability to solve problems of finding a number greater or less than a given number several times

UUD:

Cognitive:

general education –

Rightchoose an arithmetic operation (multiplication or division) for solving problems of finding a number that is several times greater or less than a given number;

call the results of all table cases of multiplication and division, as well as addition of single-digit numbers and corresponding cases of subtraction;

fulfill oral and written addition and subtraction of numbers within 100;

determine arithmetic operations for solving a variety of word problems;

realize self-monitoring of the correctness of calculations

brain teaser -

constructing reasoning in the form of a connection of simple judgments;

find different ways of solving problems;

evaluate proposed solution to the problem andjustify your assessment.

Regulatory:

take into account the rule in planning and controlling the solution method.

Communicative:

take into account different opinions and strive to coordinate different positions in cooperation.

Personal:

expand cognitive interests and learning motives;

know how to work in pairs;

understand the meaning of the boundaries of their own knowledge and “ignorance”.

Equipment:

disk “EOR for the textbook M.I. Moro. Mathematics 2nd grade";

cards for reflection;

cards for individual work, for work in pairs and groups.

During the classes

I . Motivation for learning activities

Target: inclusion of students in activities at a personally significant level:“I want because I can.”

Reflection technique “In one word”: students need to choose 3 words out of 12 that most accurately convey their state at the beginning of the lesson, and then at the end:

II . Updating and recording individual difficulties in a trial learning activity

Target: repetition of the studied material and identification of difficulties in the individual activities of each student.

Individually:

Draw relationships with arrowsmore between these numbers. Make up statements about each pair of numbers.

12 . . 23

One student on the card, the other at the board (on the back) with mutual checking:

Correct mistakes:

63: 9 = 8 (7) 3 ∙ 6 = 18 (5 + 4) ∙ 2 = 16 (18)

8 ∙ 6 = 54 (48) 45: 5 = 8 (9) 4 ∙ (8 ∙ 0) = 4 (0)

7 ∙ 4 = 28 27: 3 = 7 (9) 56: (7 ∙ 1) = 8

Verbal counting

1. How much is 15 > 5(on 10)

How many times is 15 > 5(3 times)

- How do you find out how many units one number is greater or less than another? (How many times?)

Make up questions using the numbers 7 and 28 and the word "less".

2. Insert the missing numbers and action signs:

5 * □ = 15 (+ 10; ∙ 3) 40 * □ = 5 (: 8; - 35)

9 * □ = 9 (+ 0; - 0; : 1; ∙ 1) 28 * □ = 0 (- 28; ∙ 0)

3. What is the seventh part of the number 63? a fifth of the number 35?(9; 7)

The eighth of a number is 8. Find this number.(64)

The ninth part of the number is 2. Find this number.(18)

4. A pie costs 6 rubles, and a bun is 3 rubles more expensive. How much money should you pay for a bun?(9 rubles) *Mom bought 2 buns with poppy seeds and cottage cheese and one pie. How much money did mom pay? (42 rubles) *Mom paid with a 50 ruble bill. How much change did mom get? (8 rubles) What if mom pays with a 100 ruble bill? (58 rubles)

5*. They say that point B lies on the line between points A and C if moving along this line from A to C (or from C to A) we will definitely pass through point B. This situation is shown in Figure 1.

Draw a line through points M, K, P, shown in Figure 2, so that point P lies on it between points M and K.

What is a statement? What statements about pairs of numbers have you made?

III . Inclusion in the knowledge system and repetition

The first verse of the song “It will happen again...” sounds.

How can our lesson and this song be connected?(Perhaps we will solve very difficult problems. The topic of our lesson: “Problem Solving”...)

Why do you need to be able to solve problems? How can this be useful to you in life?

Today we have a general lesson. What knowledge do we need?(We need to know what it means to increase and decrease a number several times. How to compare numbers. Multiplication and division tables...)

At computer:

Numbers from 1 to 100. Multiplication and division

Problems to find a product

Exercise 1; task 2

Perimeter of a rectangle

Task 2

1. Frontal work: solving a problem with extra data; changing the question - notebook p. 36 No. 7.

2. Work in pairs (on cards)

Write down the expressions and find their meaning:

Reduce the sum of numbers 20 and 12 by 4 times( 20 + 12) : 4 = 8

Increase the difference between numbers 11 and 9 by 8 times(11 – 9) ∙ 8 = 16

Reduce the product of 5 and 8 by 45 ∙ 8 – 4 = 36

How much is the sum of the numbers 6 and 3 greater than the quotient of the same numbers?

(6 + 3) – (6: 3) = 9 – 2 = 7

Checking visual signal plots using pictures: students make up expressions, find the answer in the pictures and lay out the desired figure in front of them.

What do these figures have in common?(These are polygons; flat figures)

In the group (Polina, Kolya, Lera, Sasha M.) they work according to the card:

1) 60: 30 = 2 (times)

2) 6: 2 = 3

Answer: 3 kg.

2. Independent solution of problems of different levels of complexity(tasks are written in different colors on one card)

Option 1:

There are 45 cars in the parking lot, and 9 times fewer trucks. How many trucks are there in the parking lot?

There are 45 cars in the parking lot, and 9 times fewer trucks. How many trucks and cars are there in the parking lot?

They brought 64 kg of cabbage to the canteen, and one-time less beets. How many kilograms of beets were brought to the canteen?

How many vegetables did you bring to the dining room?

Write expressions to solve this problem.

Option 2:

One box contained 5 kg of pears, and the other contained 8 times more. How many kilograms of pears are in the second box?

One box contained 5 kg of pears, and the other contained 8 times more. How many kilograms of pears are in two boxes?

In one box there were 5 kg of pears, and in the other there were times more. A quarter of all pears were given to children. How many pears were given to the children?

Write an expression to solve this problem.

Checking high-level problems only: Students write expressions on the board.

Do you want to help a PhD candidate? Help me create a problem that can be solved by the expression: 4 ∙ a – 4

(Mom bought 4 pies with cottage cheese, and with jam there were times more. How many more pies did mom buy with jam than with cottage cheese?

Mom bought 4 pies with cottage cheese, and with jam there were times more. We ate 4 pies with jam. How many pies with jam are left?)

IV . Homework of students' choice

(54 – 46) 5 (8 3 + 4 4) : 4

(15 + 6) : 3 (6 4 – 9) : 5 8: 4

(25 + 7) : 4 (28: 7 + 52) : 8 7

71 – 15: 3 28: (7 – 3) + 81: 9

High level task:

There is a rectangle whose length is 8 cm and width 2 cm. We need to reduce the length and increase the width of this rectangle to get a square whose perimeter is equal to the perimeter of this rectangle. Which of these figures will fit more squares with a side of 1 cm?

When the father was 30 years old, the son was 5. Now the father is twice as old as the son. How old are father and son now?

V . Reflection on learning activities in the lesson (result)

Filling out the table “In one word”

Target: students’ awareness of their learning activity (learning activity), self-assessment of the results of their own and the entire class’s activities.

Continue the sentences:

I realized that...

It was interesting…

It was difficult…

I wanted…

I managed…

Correct mistakes:

63: 9 = 8 3 ∙ 6 = 18 (5 + 4) ∙ 2 = 16

8 ∙ 6 = 54 45: 5 = 8 4 ∙ (8 ∙ 0) = 4

7 ∙ 4 = 28 27: 3 = 7 56: (7 ∙ 1) = 8

12 . . 23

For 6 kg of potatoes they paid 60 rubles. How many kilograms of potatoes can you buy for 30 rubles?

____________________________________________________________________________________________________________________________________________________________________________________________________

Consider and evaluate (true or false) this way of solving the problem:

1) 60: 30 = 2 (times)

2) 6: 2 = 3

Answer: 3 kg.

1736, Koenigsberg. The Pregelya River flows through the city. There are seven bridges in the city, located as shown in the figure above. Since ancient times, the inhabitants of Königsberg have struggled with a riddle: is it possible to cross all the bridges, walking on each one only once? This problem was solved both theoretically, on paper, and in practice, on walks - passing along these very bridges. No one was able to prove that this was impossible, but no one could make such a “mysterious” walk across the bridges.

The famous mathematician Leonhard Euler managed to solve the problem. Moreover, he solved not only this specific problem, but came up with a general method for solving similar problems. When solving the problem of the Königsberg bridges, Euler proceeded as follows: he “compressed” the land into points, and “stretched” the bridges into lines. Such a figure, consisting of points and lines connecting these points, is called COUNT.

A graph is a collection of a non-empty set of vertices and connections between vertices. Circles are called vertices of the graph, lines with arrows are arcs, and lines without arrows are edges.

Types of graphs:

1. Directed graph(briefly digraph) - whose edges are assigned a direction.

2. Undirected graph is a graph in which there is no direction of the lines.

3. Weighted graph– arcs or edges have weight (additional information).

Solving problems using graphs:

Task 1.

Solution: Let us denote the scientists as the vertices of the graph and draw lines from each vertex to four other vertices. We get 10 lines, which will be considered handshakes.

Task 2.

There are 8 trees growing on the school site: apple tree, poplar, birch, rowan, oak, maple, larch and pine. Rowan is taller than larch, apple tree is taller than maple, oak is lower than birch but taller than pine, pine is taller than rowan, birch is lower than poplar, and larch is taller than apple tree. Arrange the trees from shortest to tallest.

Solution:

The vertices of the graph are trees, indicated by the first letter of the tree name. There are two relationships in this task: “to be lower” and “to be higher.” Consider the relation “to be lower” and draw arrows from a lower tree to a higher one. If the problem says that the mountain ash is taller than the larch, then we put an arrow from the larch to the mountain ash, etc. We get a graph that shows that the shortest tree is maple, followed by apple, larch, rowan, pine, oak, birch and poplar.

Task 3.

Natasha has 2 envelopes: regular and air, and 3 stamps: rectangular, square and triangular. In how many ways can Natasha choose an envelope and a stamp to send a letter?

Solution: