Applying Algorithmic Thinking to Teaching Graphs of Functions For Students Through Geogebra

Algorithmic thinking is a term that is of interest to many educators and teachers. Algorithmic thinking plays an important role not only in problem solving but also in solving real world problems. The article presents some concepts of algorithmic thinking; propose the process of applying algorithmic thinking to teaching function graphs for students through GeoGebra online, helping students to draw all functions in the fastest way. GeoGebra is integrated with algorithms used to graph any function online that students cannot do. GeoGebra is used effectively, interactively and actively supported by many students, students and teachers of Mathematics in the process of graphing functions and graphs in an intuitive and detailed way, thereby developing develop students' thinking.

application of mathematical skills to resolve daily life problems.Ayllón and colleagues (2016) pointed out the relationship between the development of mathematical thinking, creativity, and problemsolving.Regarding creativity and mathematics, Boaler and colleagues (2017) suggested that in order to integrate brain science into their math teaching, educators should be guided on the most effective techniques to overcome the concepts they need to teach.Chu Cẩm Thơ (2016) highlighted concepts and knowledge related to thinking, mathematical thinking, and provided some methods to nurture and develop thinking in students through math teaching activities.Therefore, "mathematical thinking" is a term of great interest and utility for many scientists and researchers worldwide.To gain a deeper understanding of a type of thinking in the teaching of mathematics at the high school level, after presenting the concepts of algorithm, algorithmic thinking, and giving examples, we propose the algorithmic thinking process in teaching the Graphs of Functions to students through GeoGebra -a free online tool that provides a multi-functional mathematical computation and graphing environment.

Algorithms
Concept: According to Nguyen Ba Kim (2011), an algorithm is an exact and unambiguous set of finite elementary operations performed in a specific order on objects in such a way that after each of these finite operations, the desired result is obtained.
According to Bui Van Nghi (1996), algorithms have characteristics, features, and procedures: Characteristics: It is a finite sequence of steps arranged in a specific order; Each step is a basic operation, and in some cases, it can be a known algorithm; After completing the steps, a result is achieved.
Features: Correctness, determinism, universality, and termination.Thus, we can understand that an algorithm is a system of rules to find a solution procedure for a problem.
Algorithm Procedure: A procedure is a sequence that must be followed to perform a certain task.
A procedure can be divided into steps; each step is an activity for a specific purpose, and an activity can consist of several operations.An algorithm procedure is understood as a procedure for performing the steps of a particular algorithm.

Algorithmic Thinking
According to Bui Van Nghi and colleagues (2005), algorithmic thinking is a way of thinking to perceive and solve problems in an ordered sequence (arranged sequentially, in order).According to Chu Cam Tho (2016), algorithmic thinking is a logical inference process, using reasonable steps to solve problems, including: performing operations in a specific sequence appropriate for a given algorithm; analyzing an activity into component operations performed in a specified sequence; generalizing an activity on individual objects into an activity on a class of objects; accurately describing the process of performing an activity.
Algorithmic thinking is a prerequisite for creativity.To be creative, one needs to master the knowledge and competently perform basic skills.Using algorithmic thinking helps us identify the signs of concepts, build algorithms for problems, making it easy to grasp scientific concepts and contribute to the formation of skills and artistry.
According to Kanaki and colleagues (2022), algorithmic thinking is closely related to basic skills, such as abstract reasoning, logic, structured thinking, problem-solving skills, creativity, the ability to identify the structural elements of a problem, top-down design, repetition, identifying optimal solutions, data organization, generalization, and configurational ability.Based on these principles, it can be predicted that in many cases, students perceive learning algorithms as a challenging and unattractive process.Its complexity makes algorithmic thinking a difficult skill to develop and requires the establishment of effective teaching methods to enhance that skill.
From known algorithms, we can construct solutions to various specific problems.Algorithmic thinking helps students not only solve specific problems but also think about solving a class of problems, thereby forming algorithms that are applicable to various problem classes.Discovering an algorithm for a specific problem already demonstrates creativity.Algorithmic thinking also contributes to the development of intelligence, training skills, and the best thinking for learners.

Example of Algorithmic Thinking:
Here is a simple example of algorithmic thinking to solve the problem of graphing the function y = 2x -3: Problem: Graph the function y = 2x -3 Algorithmic Solution: Step 1: Determine the slope and y-intercept.
The slope of the function y = 2x -3 is 2. The y-intercept of this function is -3.
Step 3: Graph the function based on the chosen points.
Based on the two chosen points, we can graph the function as follows: Plot the points (0, -3) and (1, -1) on the coordinate plane.
By changing the values of x and calculating the corresponding values of y, we can see that all the values of y lie on the line.Therefore, the graph is accurate.
This is an example of algorithmic thinking to solve a simple problem.In reality, more complex problems will require more creative thinking and optimizing algorithms for higher efficiency.However, fundamental algorithmic thinking lays the foundation for solving more complex problems in computer science and many other fields.

Applying Algorithmic Thinking in Teaching the "Graphs of Functions"
To apply algorithmic thinking to teach the "Graphs of Functions," let's take an example: Given y = f(x) = x 3 -3x + 2, we can guide students to graph the function using a simple and logically structured method.Here is an algorithmic approach to graph this function: Step 1: Define the objective of graphing and the characteristics of the function: First, we need to determine critical points, inflection points, x-intercepts, y-intercepts, special points, and other characteristics of the function.
Step 2: Identify the critical points of the function: The critical points of the function are the points where the derivative of the function equals zero or does not exist.In this case: The derivative of the function f(x) is f'(x) = 3x 2 -3.
To find the critical points, solve the equation f'(x) = 0: Step 3: Determine the function's intervals: The intervals of the function are the ranges where the function is increasing (monotonic) or decreasing (non-monotonic) continuously.To determine these intervals, we need to identify the increasing or decreasing direction of the function based on its derivative.In this case: When x < -1, f'(x) > 0, the function is increasing.
Step 4: Construct the graph of the function: For the x-intercepts, solve the equation f(x) = 0, which is x 3 -3x + 2 = 0, and find the corresponding x values.
For the special points, you can find points like (-2, 0) and (2, 4) to make the graph more accurate.
After determining these special points, you have enough information to graph the function y = x 3 -3x + 2. Plot these key points, and based on the increasing or decreasing direction, you can draw the shape of the graph.Then, students can follow these steps to graph the function y = x 3 -3x + 2 logically.
With algorithmic thinking, we can guide students to perform the steps of graphing the function y = x 3 -3x + 2 scientifically and logically.This approach helps students gain a better understanding of the steps involved in solving and graphing functions and helps them develop algorithmic thinking for solving other problems.

Applying Algorithmic Thinking in Teaching the "Graphs of Functions" Using the Website https://www.geogebra.org/calculator Introduction to the Website https://www.geogebra.org/calculator
The website https://www.geogebra.org/calculator is a free online tool that provides a versatile mathematical computing and graphing environment, developed by GeoGebra, an international mathematics education organization.This tool allows users to perform calculations and create mathematical graphs.
According to Borkulo and colleagues (2021), the use of GeoGebra for developing algorithmic thinking has proven effective in various studies, including: Using GeoGebra helps students understand and visualize mathematical concepts more intuitively.
Using GeoGebra helps students develop programming skills and algorithmic thinking.
Using GeoGebra helps students analyze and solve mathematical problems by employing computational tools and generalization.
Using GeoGebra enhances student engagement and active participation in learning activities.

Key features of GeoGebra include:
Graphing functions: It can graph various functions, circular diagrams, lines, parabolas, hyperbolas, and other geometrical shapes.This aids in better visualizing relationships and properties of mathematical functions.
Limit and derivative calculations: GeoGebra allows users to compute the limits of functions at specific points and calculate the derivatives of functions with respect to the independent variable.
Creating geometric objects: GeoGebra enables the creation of geometric objects such as points, line segments, circles, rectangles, triangles, and many other geometric shapes for representation and performing related calculations.
Sharing and storing: Users can save their calculations online, share them with others, and receive feedback from the GeoGebra community.
In summary, GeoGebra Calculator is a powerful and convenient online tool for performing mathematical calculations and graphing.It is suitable for students, teachers, and anyone interested in mathematics.GeoGebra is effectively used by many students, learners, and teachers in teaching mathematics, as it provides an interactive and supportive platform for graphing functions visually and in detail by simply inputting coefficients and function expressions.

Algorithmic Process for Drawing Graphs of Functions Using GeoGebra Online
To graph functions using GeoGebra, users can follow these steps: Step 1: Define the objective and access GeoGebra online: This is the first and most crucial step of the algorithmic process for drawing graphs of functions using GeoGebra online.This step helps you clearly define the problem's objective and prepare the necessary tools.
The goal of the problem is to graph a mathematical function.To graph a mathematical function, you need to specify the following: What is the function you want to graph?
What is the form of the function?
What are the characteristics of the function?
Step 2: Select the function to graph on the GeoGebra interface: Before entering the function into GeoGebra, you need to determine which function you want to graph and what types of functions GeoGebra can graph.
You can identify the functions that GeoGebra can graph, such as linear functions, quadratic functions, cubic functions, n-degree functions, trigonometric functions, logarithmic functions, etc.
Step 3: Enter the function to be graphed into the Input window: On the GeoGebra interface, click the "Input" button to enter the function you want to graph.At this step, you should input the coefficients and variables from left to right.
Step 4: Press Enter to display the function on the coordinate plane: GeoGebra will display the graph of the function you want to graph.You can change the coefficients of the function, and GeoGebra will show the corresponding new graph to stimulate thinking and exploration for students.
-Description of the Process of Drawing Function Graphs Using GeoGebra (Figure 2) Figure 2. Description of the Process of Drawing Function Graphs Using GeoGebra

Drawing Function Graphs Using GeoGebra Online
Returning to example 3 with the function f(x) = x 3 -3x + 2, applying the process of drawing function graphs using GeoGebra online as described above, we obtain the graph as shown in Figure 2: Here is the algorithmic thinking involved in drawing the graph of the function f(x) = (x 3 -3x + 2) using GeoGebra online: 1) Define the problem: Understand that the goal is to plot the graph of the function f(x) = (x 3 -3x + 2).
2) Input the function: Access the GeoGebra Calculator website and input the function f(x) = (x 3 -3x + 2) into the input bar.
3) Determine the range: Decide the x and y ranges to be displayed on the graph.For example, you can select a range from -5 to 5.
4) Create a coordinate grid: GeoGebra automatically displays a coordinate grid to determine the positions of points on the graph.
5) Calculate the points on the graph: Use the function f(x) = (x 3 -3x + 2) to calculate the corresponding y-values for each x-value within the selected range.
6) Mark the points: Use the marking tool to label the points with coordinates (x, y) on the graph.
7) Connect the points: Use the connecting tool to create a line that represents the graph.
Algorithmic thinking is manifested through the systematic approach to problem-solving, from inputting the function to calculating values, marking points, connecting them, and customizing the graph.These steps build upon the results of the previous ones and help create an accurate representation of the given function.
Algorithmic Thinking in Solving the Graphing Equations f(x) = 3 x + 4 x and g(x) = 5 x Using GeoGebra Online to Solve Non-standard Equations: 3 x + 4 x = 5 x Non-standard equation 3 x + 4 x = 5 x is an equation that cannot be solved using a common method for all equations; it depends on the specific characteristics of each equation.
Before GeoGebra Online was available for solving the equation 3 x + 4 x = 5 x , one had to guess the solution, such as x = 2, and then prove that x = 2 is the unique solution using the following method: a) The conventional way to solve equations like 3 x + 4 x = 5 x involves these steps: Step 1: Transform the equation into Step 2: Assume x = 2 as the solution for (*) because . Then, we have to prove that this is the unique solution.
Step 3: For the function 3 4 () 5 5 Step 4: Therefore, the equation has only one unique solution.
Though you can solve the equation 3 x + 4 x = 5 x using this method, not all problems can be solved this way.With GeoGebra Online, finding a unique solution through graphing is much more accessible, and then you can prove the unique solution as follows: b) Using GeoGebra Online to solve the equation 3 x + 4 x = 5 x : Step 1: Problem objective: Use GeoGebra Online to solve a non-standard equation, 3 x + 4 x = 5 x Step 2: Input functions: Access GeoGebra Online, then input the two functions f(x) = 3 x + 4 x and g(x) =5 x into GeoGebra's data input window.
Step 3: Find the intersection point of the two graphs of f(x) = 3 x + 4 x and g(x) =5 x on GeoGebra's interface to determine the solution of the equation 3 x + 4 x = 5 x , as shown in Figure 4. Step 4: Determine the solution: Based on the intersection point of the two graphs, f(x) = 3 x + 4 x and g(x) =5 x .intersect at x = 2.This means that x = 2 is the solution to the equation 3 x + 4 x = 5 x .

Figure 4 .
Figure 4. Intersection of the two graphs of f(x) = 3 x + 4 x and g(x) =5 x 's a decreasing function over R because