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Teacher activity

Student activities

Organizing time

Good day! In the previous lessons, we got acquainted with logical operations and their truth tables. And today we will get acquainted with solving problems in several ways.

Greet the teachers.

Preparation for the perception of new material.

In everyday life, we all have to deal with big and small logical problems. The solution of logical problems to a certain extent simulates the solution of scientific problems. Professional human activity in almost all industries requires the ability to build logical models, analyze a lot of disjointed data, facts, draw conclusions, conclusions. Logic problems stand apart in the vast kingdom of problems that are solved at school. They do not require special knowledge, be it mathematics or physics. In them there is no play on words, no attempts to mislead a person. Logic problems are very diverse and their solution cannot be reduced to one or two standard schemes.

Listen to the teacher.

Explanation of the new material

Let's look at the problem (slide 2).

Objective 1. When drawing up the schedule for Tuesday in grade 11, the teachers made requests to the head teacher. Math teacher: "I wish you had the first or second lesson." History teacher: "I wish you had the first or third lesson." Literature teacher: "I wish you had a second or third lesson." What schedule will be drawn up if there can be only one lesson for each subject.

How do you think this problem can be solved?

Right. How then do we start solving the problem in this way?

Right. Let's solve this problem on the board. Anyone interested?

In this method we use "Electronic pen". It allows you to fill out the truth table (slide 3).

How many lines will the truth table consist of?

Guys, do you think this is a rational way to solve the problem? Why?

Absolutely right. This problem can be solved in several ways. Let's consider them.

2nd method (tabular)

The tabular method for solving logical problems is simple and clear, but it can be used only when it is required to establish a correspondence between two sets. Tell me, can we use this method when solving a problem?

To solve the problem in a tabular way, you need to know the following rules:

1. Each row and each column of the table can contain one correspondence sign (for example, "+").

2. If in a row (or column) all the “places”, except one, are occupied by an elementary prohibition (an inconsistency sign, for example, “-”), then a “+” sign should be put in the empty space; if the row (or column) already has a "+" sign, then all other places must be occupied by a "-" sign.

Thus, the solution will be completed when we are able to place one plus in each row and column, thus indicating the schedule for each subject.

Before we start we need to build a table. How many columns and rows will it have?

And now we start solving the problem using an interactive whiteboard (slide 4). Guys, we see that we have the names of columns and rows, exactly the opposite. Tell me, is it fundamentally important? What do we know by condition?

This means that in the table at the intersection of these names we put "+", we will pay attention that in one statement we have two conditions. Therefore, the first condition is denoted by"+", and the second is "+". And in the same way, we check the second and third statements.

As a result, we come to what conclusion?

Answer: mathematics - 1 lesson, literature - 2 lesson, history - 3 lesson or history - 1 lesson, mathematics - 2 lesson, literature - 3 lesson.

There is another way to solve logical problems.

3rd way (columns)

The concept of a graph is not new for you, so we will not dwell on the definition in detail.

Graph Is a collection of objects with connections between them. Objects are represented as vertices or nodes of the graph, and links as arcs or edges. If a unidirectional relationship is indicated on the diagram by lines with arrows, if a two-way relationship between objects is indicated on the diagram by lines without arrows.

Solution: according to the condition of the problem, let's make a graph, guys, what do you think, the vertices of this graph will be, what? Well done. What will the solid line represent?

It is clear from the problem statement that mathematics will be the first lesson or the second. On the board you needestablish correspondence (move statements to the required designations) (slide 5).

Then we check the second condition: the story is either the first lesson on schedule, or the third lesson. And the third condition: literature - either the second lesson on schedule, or the third lesson.

Looking at the graph, what conclusion can be drawn?

And the last way we will get acquainted is the 4th way (simplifying the logical expression)

Under Simplification of Boolean Expressions understand an equivalent transformation that leads to a formula that either contains, compared to the original, fewer operations and does not contain negations of non-elementary formulas, or contains a smaller number of occurrences of variables.

Some logical formula transformations are similar to formula transformations in regular algebra. Name these transformations.

Whereas other transformations are based on properties that ordinary algebra operations do not have. Let's consider the solution to the problem. First you need to highlight simple statements and designate a certain letter.

You have already done this task today, name them.

You must make a logical formula from the proposed list of designations (by dragging the desired symbol into place) (slide 6).

And then, let's compose an expression.

There are those who wish to the board.

This expression was simplified using the laws of logic (slide 7).

What do you see? (Pupils come to the conclusion independently, by reasoning)

Guys, we have analyzed 4 ways to solve one logical problem, tell me which way is the most rational in your opinion.

Listen carefully to the problem statement.

This problem can be solved using a truth table.

It is necessary to highlight simple statements and designate them with a specific letter. Then write down the formula, and then draw up a truth table. Yes. Comes out and solves the problem.

M1 - "1 lesson of mathematics"

M2 - "2 lesson of mathematics"

I1 - "1 lesson of history"

I3 - "3 history lesson"

L2 - "2 literature lesson"

L3 - "3 literature lesson".

Make up the expression (M1 + M2) * (I1 + I3) * (L2 + A3)

The truth table will consist of 64 lines.

Not. Since there are a very large number of lines, it will take us a lot of time to solve the problem.

Write down method 2 in a notebook.

Yes, since we need to establish a match.

The willing student goes to the blackboard.

The table consists of three rows and three columns. The columns will indicate the name of the item, and the rows will be the number in order.

No, it doesn't matter.

By hypothesis, we know that the math teacher wants his lesson to be the first or second,

In this example, students used an element of interactivity such as “Magnifying glass” - to fill in the table (when you hover the “magnifying glass” over the required cell, they get the correct answer).

As a result, we come to the conclusionthat mathematics - 1 lesson, literature - 2 lesson, history - 3 lesson or history - 1 lesson, mathematics - 2 lesson, literature - 3 lesson.

Write down the answer in a notebook.

The third method and definition are recorded.

The tops will be the name of the items and the serial number of the item.

A solid line will indicate that the item corresponds to a certain serial number of the item.

Complete the task.

Looking at the graph (we can conclude that mathematics - 1 lesson, literature - 2 lesson, history - 3 lesson or history - 1 lesson, mathematics - 2 lesson, literature - 3 lesson.

Write down the fourth method.

Taking the common factor out of the brackets, using the travel and combination laws, etc.),

Simple sayings:

M1 - "1 lesson of mathematics"

M2 - "2 lesson of mathematics"

I1 - "1 lesson of history"

I3 - "3 history lesson"

L2 - "2 literature lesson"

L3 - "3 literature lesson"

Yes, they go to the board and decide.

As a result, we got a logical expression in which we see that mathematics - 1 lesson, literature - 2 lesson, history - 3 lesson or history - 1 lesson, mathematics - 2 lesson, literature - 3 lesson.

Opinions about most rational way parted.

Lesson summary

Do you think logic problem solving develops logic?

Right. Solving logical problems to a certain extent simulates the solution of scientific problems. A person's professional activity in almost all branches of knowledge requires the ability to build logical models, analyze a lot of disjointed data, facts, draw conclusions, conclusions. The conclusion we have come to is a unified approach to solving problems, both educational and scientific. By putting forward hypotheses and consistently reasoning, formulating conclusions and examining their compatibility with the initial data, the researcher, in the end, gets a certain exact conclusion, having at first a lot of more or less disjointed data.

And now guys, next to the tasks that we solved, you must put the symbol for a difficult task - an asterisk, medium difficulty- circle, light - triangle.

Yes, when solving problems, you need to show ingenuity, ingenuity, you need to analyze, compare, draw conclusions.

Complete the task.

Homework. Come up with a logical problem and solve it. Make a solution in the form of a presentation.

Grading.

Write down homework.