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10DIFFICULTIES OF THE TOPIC «QUADRATIC FUNCTIONS AND CONSTRUCTING THEIR DRAWINGS» AND WAYS TO OVERCOME THEM FOR
STUDENTS
A.K. ISKAKOVA
Almaty University of Power Engineering and Telecommunications named after
G.Daukeev.
A.T. SHILDEBAY
Kazakh National Women's Pedagogical University.
Abstract. This article explores the methods andprinciples of Lesson Study in the context of teaching mathematics, with a focus on quadratic functions and graphing. It discusses the main challenges that students face when studying this topic, such as the complexity of the material, insufficient motivation, and difficulties in differentiated instruction. The article describes ways to overcome these challenges through individualized approaches, teacher support, and the use of various teaching methods, including Lesson Study. Examples of task classification by level of difficulty, type of activity, and cognitive task type are provided, along with detailed solution methods for examples. The presented material is of interest to both teachers and researchers in the field of education, aiming to develop more effective teaching strategies and understanding of mathematical concepts.
Keywords: Lesson Study, skills, abilities, quadratic functions, classification.
Introduction. Throughout history, two main purposes have shaped mathematical education: practical, linked to the creation and application of tools necessary for human productivity, and intellectual, associated with human thought, mastery of specific methods of cognition, and transformation of reality through mathematical methods. The practical utility of mathematics is grounded in its focus on fundamental structures of the real world: spatial forms and quantitative relationships - from the simplest, grasped through direct human experience, to the complex, necessary for the development of scientific and technological ideas. Without concrete mathematical knowledge, understanding the principles of modern technology, perceiving various types of information, and everyday practical activities become less efficient.
Methodology. Without a basic mathematical foundation, modern education cannot be established. In schools, mathematics serves as a foundational subject for the study of related disciplines. An increasing number of fields requiring high levels of education are directly tied to the application of mathematics [1]. Thus, the circle of students for whom mathematics becomes a professionally significant subject is expanding.
Teaching mathematics using Lesson Study in schools not only serves general educational goals but also has specific aims:
1. Creating a safe and supportive environment where errors are seen as part of the learning process is crucial. Students feel more comfortable when they know they can make mistakes and learn from them.
2. Impact on students. Implementing Lesson Study when teaching quadratic functions expands students' opportunities in several ways: improving problem-solving skills (students acquire skills in applying various problem-solving strategies, which strengthens their confidence in solving complex mathematical problems); fostering critical thinking (students develop critical thinking skills by analyzing, evaluating, and applying different methods to understand quadratic functions); enhancing self-esteem (overcoming difficulties within Lesson
Study contributes to a sense of accomplishment, enhances students' self-esteem, and motivates them to solve more challenging problems).
In addition to the aforementioned objectives, Lesson Study in mathematics education is also aimed at stimulating collective learning. Collaboration among teachers enables the exchange of ideas, experiences, and best practices, fostering an atmosphere of mutual assistance and support. This approach significantly contributes to the professional development of teachers and enhances their effectiveness. Lesson Study also creates space for reflection and self-assessment. Teachers and students can collectively analyze lessons, identify their strengths and weaknesses, and discuss ways to improve. This promotes a deeper understanding of the learning process and develops skills in reflection and self-assessment.
The use of Lesson Study opens up new opportunities for integrating modern technologies into teaching. Contemporary tools such as interactive whiteboards, online resources, and educational applications can enhance the effectiveness of instruction and make it more interactive and engaging for students. Working in groups within Lesson Study contributes to the development of students' communication skills. They learn to express their thoughts, articulate their viewpoints, and listen to others' opinions, which is beneficial both in their studies and in life. Finally, Lesson Study allows for the application of differentiation principles, taking into account individual students' needs. This promotes more effective adaptive learning and accommodates the diversity of students' levels of preparation and abilities.
Given the significant variability in students' individual characteristics and class sizes, teachers often struggle to adequately address each student's needs, leading to a focus on the average student who may feel somewhat comfortable with such instruction. However, teachers must always consider students' individual characteristics. Alongside this, teachers may encounter several difficulties, as illustrated by the topic of "Quadratic Functions and Graphing," along with possible ways to overcome them:
1. Complexity of material and difficulties in explaining concepts: The topic of quadratic functions can be challenging for students due to the abstract nature of their definitions and properties, such as the vertex of a parabola or the concept of discriminant. Teachers may face challenges in explaining the material and capturing students' attention. Overcoming this: Using visual examples such as graphs, diagrams, and interactive applications to visualize quadratic functions can help students better understand the material. It is also helpful to relate the material to real-life examples to make it more accessible.
2. Lack of motivation among students: Some students may not see the practical applications of quadratic functions in real life and therefore may not show interest in the topic. Overcoming this: Demonstrating specific examples where quadratic functions are used, such as in physics to describe motion or in economics to model production costs.
3. Difficulties in differentiating instruction: Classes may have students with varying levels of preparedness, making it challenging to approach teaching that accommodates their differences. Overcoming this: Using differentiated assignments and teaching methods such as group work, individual consultations, and additional materials for self-study helps tailor instruction to students' diverse needs.
4. Insufficient teacher preparation: Some teachers may lack the knowledge or experience in teaching the topic of quadratic functions. Overcoming this: Providing additional training and educational programs for teachers, as well as sharing experiences with colleagues, contributes to enhancing their qualifications and confidence in teaching the topic.
5. Insufficient technology usage skills: Some teachers may use outdated or ineffective teaching methods, hindering students' understanding of the material. Overcoming this: Applying modern teaching methods such as Lesson Study, where teachers collaborate on developing and improving lessons, can help enhance the effectiveness of instruction.
Additionally, conducting training and seminars on using technology in education and exchanging experiences with more experienced colleagues helps teachers master new teaching methods and tools.
Problem-solving plays a significant role in mathematical education. Psychological studies on the issue of teaching problem-solving indicate that the main reasons for students' lack of general skills and abilities in problem-solving stem from insufficient knowledge of the essence of problems and their solutions. As a result, students often solve problems without properly understanding their own activity. Moreover, students may struggle to develop separate skills and abilities required for problem-solving actions, thus needing to acquire these skills during the problem-solving process, which many find challenging [2]. Quadratic functions often pose difficulties for students due to their abstract nature. Understanding the given equation of a parabola, reducing the equation to its canonical form, and comprehending the location of vertices on the graph can be challenging tasks. Students can easily feel disappointed when encountering these complex mathematical concepts.
Let's consider the difficulties of the topic "Quadratic Functions and Graphing" and ways to overcome them for students. The topic "Quadratic Functions and Graphing" can be challenging for students for several reasons. Here are some possible difficulties and ways to overcome them:
1. Understanding the concept of a quadratic function: Students may struggle to grasp what a quadratic function is and how it relates to the graph. Overcoming this challenge involves using visual examples, graph demonstrations, and interactive applications to illustrate how the graph changes depending on the function's parameters.
2. Graphing a quadratic function: Students may have difficulty graphing a quadratic function and understanding its shape. This difficulty can be addressed by explaining the main characteristics of the graph of a quadratic function (vertex, direction of branches, axis of symmetry) and conducting practical exercises on graphing.
3. Solving quadratic function equations: Students may encounter difficulties in solving equations related to quadratic functions. To overcome this challenge, step-by-step explanations of the equation-solving process, the use of examples with different types of equations, and practical exercises to reinforce skills are recommended.
4. Applying quadratic functions in problems: Students may struggle with applying quadratic functions in various problem-solving contexts. Providing diverse problem-solving tasks involving quadratic functions in real-life situations can help students recognize the practical value of these functions.
Quadratic functions and graphing represent one of the key topics in the mathematics curriculum. However, students often encounter various difficulties when studying this subject. In addition to the difficulties already mentioned, there are other aspects that require special attention from teachers and curriculum developers.
Understanding the precision and interpretation of graphs: For many students, adequate reading and proper interpretation of the graph of a quadratic function can be challenging. They may have difficulty determining characteristics of the graph, such as the vertex, direction of branches, and points of intersection with the axes. This can lead to incorrect conclusions and inaccurate problem-solving.
Example: Consider the function y = x2- 4x + 4 [7, p. 10]. Its graph can be interpreted as an upward-facing parabola with the vertex at the point (2; 0). Such understanding allows students to analyze the behavior of the function correctly and use this information to solve problems.
Difficulties with solving systems of equations: Solving systems of equations containing quadratic functions can also be challenging for students. The need to apply various solution methods, such as substitution or elimination, can complicate the problem-solving process.
( y = X2
Example: Consider the system of equations { _ 2 + 4/ Students need to find the
points of intersection of the graphs of both functions, which requires the ability to solve quadratic equations and analyze their geometric meaning. Understanding the connection to other mathematical concepts: For a complete understanding of the topic of quadratic functions, students must understand the connection of this topic to other mathematical concepts, such as geometry and algebra. Lack of such understanding can hinder successful mastery of the material.
Example: Understanding that a quadratic function describes a parabola on the coordinate plane requires students to understand the geometric meaning of the coefficients of this function, such as the vertex and the direction of the opening of the parabola. Ineffective use of technology: In modern education, computer programs and calculators are widely used for graphing quadratic functions and solving equations. However, insufficient understanding of the principles of such technologies can be a barrier to fully mastering the topic.
Example: Students may encounter difficulties in using software for graphing functions due to lack of experience or understanding of the basic commands and functions of the program.
Using individualized approaches, teacher support, and a variety of teaching methods can significantly improve students' understanding and mastery of the topic "Quadratic Functions and Graphing." Teachers have the ability to choose the optimal course of the lesson, which will contribute not only to the solid assimilation of new knowledge but also to the development of problem-solving skills. The main principle here is to teach students an approach to the problem where the task becomes an object of careful study, and its solution becomes an object of construction and invention. To achieve this, tasks can be classified when studying the topic "Quadratic Functions and Graphing." In Lesson Study, tasks can be classified according to several criteria, such as the level of difficulty, type of activity, type of cognitive task, etc. Here are examples of classifying tasks using the Lesson Study methodology for studying the topic "Quadratic Functions and Graphing":
1. By level of difficulty. Simple tasks: tasks requiring the application of basic formulas and rules for working with quadratic functions. For example, finding the vertex of the graph of a quadratic function, solving a quadratic equation by trial and error. Medium difficulty tasks: tasks that require a deeper understanding and application of quadratic functions. For example, graphing a quadratic function given certain points. Complex tasks: tasks that present a challenge for students and require analytical thinking. For example, solving a system of equations involving quadratic functions, finding the range of values for which the quadratic function is positive, considering an optimization problem using a quadratic function, applying knowledge of quadratic functions to solve a problem about the motion of an object.
2. By type of activity. Research tasks - tasks involving the analysis of graphs of quadratic functions and identifying their properties, for example, comparing the graphs of two quadratic functions and describing how changes in coefficients affect the shape of the graph. Creative tasks - tasks that require creating their own quadratic functions with specific characteristics or graphing with given conditions, for example, asking students to create a problem that requires the application of a quadratic function to solve a real-world problem.
3. By type of cognitive task. Understanding - tasks aimed at understanding basic concepts and properties of quadratic functions. For example, explaining how to determine the direction of the branches of the graph of a quadratic function; explaining how changes in coefficients affect the position of the graph of a quadratic function. Application - tasks where students must apply knowledge of quadratic functions to solve practical problems. For example, solving a maximum/minimum problem using a quadratic function [5].
Example of a task classified by level of difficulty - simple task: find the vertex of the graph of the function y = x2 — 2x — 3 [6, p. 19]. Example of a task classified by type of activity - research task: compare the graphs of the functions y = x2and y = —x2 — 4. Example of a task classified by type of cognitive task. Task: explain how changing the coefficient an affects the shape of the graph of the quadratic function y = ax2 + ¿x + c. Classification: understanding.
Examples of task classification for students studying the topic "Quadratic Functions and Their Graphs":
1. Investigative tasks. Task: Compare the graphs of the functions y = x2 and y = —x2 + 4. What can be said about the effect of the coefficient before x2 on the shape of the graph?
To complete this task, we will first plot the graphs of both functions and then compare
them.
Solution: Let's plot the graph of the function y = x2. To do this, we need to know the values of the function y for different values of x. Let's create a table of values for y = x2:
X -2 -1 0 1 2
y 4 1 0 1 4
Now, let's plot these points on the coordinate plane and draw the graph of y = x2. Let's plot the graph of the function y = —x2 + 4. To do this, we'll also construct a table of values:
X -2 -1 0 1 2
y 0 3 4 3 0
Now, let's plot the graph by marking the points with coordinates from the table on the coordinate plane.
Comparison of graphs:
The graph of the function y = x2 represents an upward-opening parabola with the vertex at the point (0,0).
The graph of the function y = —x2 +4 is also a parabola, but it opens downwards and is shifted upwards by 4 units relative to the y-axis.
The influence of the coefficient before x2 lies in the direction of the parabola's opening and its vertical position on the coordinate plane.
Task: Plot the graph of the quadratic function y = x2 — 4x + 3 and determine its vertex [6, p. 54].
2. Creative Task: Propose your own example of a quadratic function and plot its graph to demonstrate a specific property of quadratic functions.
Suppose we want to demonstrate the property of compression/stretching of the parabola along the x-axis when changing the coefficient a in the equation y = ax2. For example, let's consider the quadratic functions y = 2x2 and y = — ^x2 [3, p. 103].
Plotting the graph of the function y = 2x2:
This function represents a parabola opening upwards with a compression coefficient
a = 2.
We can plot the graph using the methods described earlier and display the parabola on the coordinate plane.
i 7
Plotting the graph of the function y = —
This function also represents a parabola, but with a compression coefficient a = --,
which makes the parabola wider than in the first case.
Let's plot this graph and compare it with the previous one.
Task: Students are assigned the task of applying a quadratic function to solve the following practical problem: A company manufactures and sells a product. The total cost (in dollars) of producing x units of the product is given by the quadratic function C(x) = 0,1x2- 20x + 500 ), where C(x) represents the total cost. Determine the number of units that need to be produced to minimize the total cost.
3. Understanding Task. Task: Explain how changes in the coefficients affect the shape of the graph of a quadratic function.
Solution:
The quadratic function has a general form of y = ax2 + bx + c, where a, b and c are the coefficients of the function.
The coefficient a determines the direction and degree of opening of the parabola:
- If a > 0, the parabola opens upwards, and its degree of opening depends on the magnitude of I a I. The larger the magnitude of I a I, the narrower and more elongated the parabola.
- If a < 0, the parabola opens downwards, and its degree of opening also depends on the magnitude of I a I.
- The coefficient b affects the horizontal shift of the parabola along the x-axis and determines the location of the vertex of the parabola. A positive value of b shifts the parabola to the left, while a negative value of b shifts it to the right.
- The coefficient c determines the vertical shift of the parabola along the y-axis. If c > 0, the parabola shifts upwards, and if c < 0, the parabola shifts downwards.
Therefore, changing the values of the coefficients a, b and c leads to changes in the shape and position of the graph of the quadratic function on the coordinate plane.
Task: How can you determine the direction of the branches of the graph of a quadratic function?
Solution:
The direction of the branches of the graph of a quadratic function (parabola) depends on the sign of the coefficient a in the equation of the function y = ax2 + bx + c:
- If the coefficient a > 0, then the branches of the parabola are directed upwards.
- If the coefficient a < 0, then the branches of the parabola are directed downwards.
4. Application Tasks:
Task 1: Solve the equation —x2 — 6x + 5 = 0 using the discriminant and find the points of intersection with the coordinate axes [4, p. 116].
Task 2: Find the maximum or minimum value of the function y = 2x2 — 6x + 5 [4, p.
116].
Within the framework of the school algebra course, problems involving quadratic functions can be classified as follows [3, 4]:
Solving Quadratic Equations: Finding the roots of quadratic equations, both by factoring and using the quadratic formula; Solving systems of quadratic equations.
Graphical Representation: Graphing quadratic functions; Determining the vertex of the parabola and the direction of its branches.
Analytical Problems: Determining extrema of quadratic functions and their values; Solving problems to find the maximum or minimum of a function within a specific context; Finding function values for specific argument values.
Interpretation of Graphs: Analyzing the behavior of the graph of a quadratic function (e.g., whether the parabola opens upwards or downwards); Determining intersections of the graph with the coordinate axes and their values.
Practical Problems: Solving problems to model real-life situations using quadratic functions (e.g., projectile motion, determining maximum height reached, time of flight, etc.). Additional Topics: Formulating quadratic equations based on given conditions; Using the discriminant to analyze the number and nature of roots of an equation.
Summery. The criterion for successful teaching should be the quality of students' mathematical preparation, the fulfillment of educational and upbringing tasks, rather than the formal use of any method, technique, form, or means of teaching. The Lessons Study system provides a structured approach to teaching quadratic functions, while simultaneously fostering students' independence in problem-solving.
One of the key aspects of the Lesson Study method is collaborative lesson study. In this process, teachers collectively analyze, plan, and conduct lessons, focusing on achieving specific learning goals. By applying the Lesson Study method to the classification of educational tasks, teachers can systematize the curriculum based on various criteria and characteristics, which promotes a deeper understanding of the curriculum and more effective student learning.
The process of classifying tasks using the Lesson Study method typically begins with an analysis of educational standards or curricula to identify the key topics and concepts that need to be taught. Then, teachers work together to identify various types of tasks that may be related to these topics. For example, in the context of quadratic functions, tasks can be classified according to the type of graph, method of solution, or word problems in which these functions are used. By applying collaboration, reflection, and adaptability, educators can create an environment in which students not only acquire mathematical concepts but also develop resilience, confidence, and a lifelong love for learning.
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