Methodical recommendations for the design and solution of problems in geometry
Problem solving is one of the main stages in the assimilation of the system of mathematical knowledge by students, in particular, geometric concepts and connections between them. Solving geometric problems, students develop creativity, independent thinking, acquire skills in the practical application of the theoretical provisions of geometry. Pedagogical practice shows that solving problems that are not united by general methods does not give good results and causes great difficulties for students.
When planning a lesson, the teacher needs to draw the students’ attention to the theoretical material necessary for solving the problem, to rethink its content in practice. Such a methodological technique will prepare students for successful perception and comprehension of a specific problem, for the conscious application of theory in practice, will help consolidate the previously studied material, the acquired mathematical knowledge will become more solid.
When solving a mathematical problem, the following stages can be distinguished:
· studying the condition of the problem;
· analysis of the solution to the problem (search for solutions);
· choosing the best way to solve the problem;
· the solution of the problem;
· research of the obtained result.
Students often omit the last step of the given algorithm, which leads to an incorrect result.
When solving geometric problems in the first lessons, it is advisable to offer students the following solution algorithm:
· Examine the condition of the problem. Sketch a drawing that meets the condition of this task.
· Understand what you need to find in the problem and what you need to know for this.
· From the system of reference tasks, select frequently repeated tasks (preferably accompanied by an illustration) that will be included in the course of solving this task.
· Find out which of the previously studied problems can be useful in solving this problem.
· Considering the previous step, reformulate this task. Try to solve it.
Knowing the system of support tasks, the teacher clearly plans the need to use a specific support task in solving this problem. This makes it possible to teach students how to “decompose” a complex problem into simpler components of the problem.
Analytical and synthetic methods are used in solving problems.
With the analytical method of solving problems, students must clearly understand that analysis is that reasoning is conducted from the sought-after to the data. Leading question — “What do you need to know to answer the main question of the problem?”. When analyzing the problem, it is necessary to draw the students’ attention to the fact that sometimes the conditions of the problem give a hint to the next leading question. With the synthetic method of solving problems, students should understand that synthetic reasoning is reasoning with the subsequent transition (using logical inferences) from the given conditions of the problem to its conclusion. The leading question in this case is “What can we learn from the given conditions of the problem?”.
Let us show how analysis and synthesis are applied using a specific problem as an example.
The ОВ beam divides the angle АС, the degree measure of which is 108о, into 2 parts. Find the degree measure of the angle BOA if the angle BOA is three times the angle ∠BOC.
We solve this problem like the previous one. Let the degree measure of the angle BOC is equal to xo, then the measure of the angle ∠BOA is equal to (3x) о. Since their sum will be equal to 108 ° (by condition), we compose and solve the equation:
X + 3x = 108
4x = 108
x = 27
Accordingly, the measure of the angle ∠BOA is equal to (3x) о, that is, 81о.
The figure shows the unfolded angle ∠АОD. The angles BOA and ∠СОD are equal. Indicate if there are still equal angles in the picture?
Consider the angles AOC and ∠BOD. They consist of equal parts ∠BOA and ∠СОD, as well as a common part ∠VOS.
Since ∠BOC is general, and ∠AOB = ∠COD (by condition), then ∠AOC = ∠BOD.
Answer: ∠AOC = ∠BOD.
Solving the problem with the help of synthetic reasoning, we sequentially select the data pairs that are dependent on each other and are necessary for solving the problem in the condition of values, we check each time whether we are approaching the final result when composing certain combinations.
Having considered the solution of the problem by the two proposed methods, students should be drawn to the fact that in the sequence of the chain of inferences, analysis and synthesis are inseparably linked. So, when analyzing, i.e. following from the question of the problem, it is necessary to pay attention to the fact that sometimes these conditions of the problem presuppose an answer to the next leading question. Conversely, following the synthetic method, i.e. By combining these tasks, we constantly keep in mind the question that needs to be answered. Thus, it follows that the organic combination of analysis and synthesis in solving problems is a unified analytical-synthetic method.