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Pham Minh Chau, Truong Thi Khanh Chi, Le Thi Thoa
Course 47, Faculty of Maths, Hanoi Pedagogical University 2
USING GEOGEBRA SOFTWARE IN TEACHING SOME CONCEPTS ABOUT STATISTICS
AND PROBABILITY GRADE 10
Abstract
Statistics and Probability is an abstract content in the 2018 General Education Program. Therefore, most teachers and students encounter difficulties in the process of teaching and learning Statistics and Probability content. This article proposes a method to help students more easily access the topic of Statistics and Probability: Teaching some concepts of Statistics and Probability in Grade 10 Math with the support of GeoGebra software. This method not only improves the quality of teaching but also arouses students' interest in the subject.
1. Introduction
The content of Statistics and Probability is very close to real life and is applied in many fields, helping learners gain a new perspective on science, with the potential for learners to comprehensively develop their qualities and abilities needed today. However, teaching Statistics and probability content to students, especially at the high school level, will have many difficulties, partly because traditional teaching methods are no longer suitable. Only studying theory in books and with it a lot of knowledge can easily cause depression and loss of concentration. The appropriate use of information technology allows students to access statistical concepts easily. GeoGebra is dynamic math software that is attracting a lot of interest in the math community. Spreadsheets that allow for statistical and probability calculations are unique features of GeoGebra, not found in other dynamic mathematics software. Therefore, using GeoGebra in teaching 10th grade Statistics and probability concepts has a positive and effective impact on students' access to knowledge.
2. Research results
2.1. Teaching Statistics and probability with the support of information technology
Teaching and learning using information technology has many advantages such as bringing more effective learning opportunities to students (Roberts, 2012); increase student engagement (White, 2012) and encourage learning exploration (Bennet, 1999). Mills (2002) proposed the use of computers in teaching probability, as a way to improve students' ability to understand abstract or difficult concepts. Batanero (2007) emphasizes that the introduction of computers into schools that enable simulations can help students solve simple probability problems, a feature that cannot be accomplished by physical experiments. Lane and Peres (2006) discuss research findings that confirm the benefits of randomized experimentation and computer simulations in developing active learning approaches that allow students to construct knowledge.
Pratt and Kapadia (2010) point out that researchers in probability cognition agree that students should be given opportunities to work with random generators as well as computer simulations, as this allows them explores two complementary types of mathematical activities: theoretical and experimental. Theoretical operations include classical combinatorial analysis obtained from experience with random generators, and experimental operations include routine testing, either by hand or by computer.
In fact, there are currently many types of technology available for teachers to teach statistical content. While the impact of technology on the practice of statistics is undeniable, the impact of technology on statistics pedagogy and recommended practices is equally powerful. For example, the Guidelines for Assessment and Instruction in Statistics Education (GAISE) framework states that "the advances in technology
and modern data analysis methods of the 1980s, along with the about society's data in the information age, has led to the development of teaching materials aimed at integrating statistical concepts into school curricula as early as the elementary grades" (Franklin & Garfield 2006). Similarly, the GAISE College Report directly recommends the use of technology to develop an understanding of statistical concepts and data analysis in the teaching of an introductory, undergraduate statistics course (Franklin & Garfield 2006). Technology has led to many changes in statistical practice. Many problems that were previously difficult to solve using analytical methods now have approximate solutions. Many of the assumptions that have been made so that statistical models can be simplified and usable no longer need to be made.
Teachers are encouraged to see the use of technology not just as a way to crunch numbers but also as a way to explore concepts and ideas and enhance student learning (Friel 2007; Garfield, Chance, & Snell 2000). Furthermore, technology should not be used just for the sake of using technology (e.g., entering 100 numbers into a graphing calculator and calculating statistical summaries) or to achieve false precision (performing result to a meaningless number of decimal places) (Franklin & Garfield 2006).
However, through practical research, we found that using information technology in teaching the topic Statistics and probability still has some difficulties such as: there are still some teachers who do not have the skills to use it. proficient in information technology, confused in teaching, so they often ignore or do not fully exploit the value of information technology in teaching Statistics and Probability. Therefore, how to effectively use information technology in teaching Statistics and Probability topics in grade 10 is an issue that needs attention and direction. In this study, we present opportunities in teaching some of the concepts of Statistics and probability that technology offers, to provide rich learning experiences for students. 2.2. Content of Statistics and Probability in grade 10
The 2018 Math general education program has presented the contents and requirements to be met for the Statistics and probability circuit in grade 10 specifically. However, teachers also realize that this content is relatively abstract and difficult for students to access. Therefore, teaching the content of statistics and probability for grade 10 needs to have a clear determination of methods, ensuring effectiveness in teaching. Information technology has become an effective tool to support teachers and students in overcoming these difficulties. The opportunities that information technology brings to help students meet the requirements of the program content are shown in the following table:
Content Requirements Opportunities to use information technology
Statistics Analyze and process data Typical numbers measure central tendency for ungrouped data samples - Calculate the number of features measuring central tendency for ungrouped data samples: average number. Provides tools to define the average.
Probability Concepts of probability Some concepts of classical probability. - Recognize some concepts of classical probability: random experiment; sample space; random events; Classical definition of probability. - Describe the sample space and events in some simple experiments (for example, rolling a dice five times). Use data visualization tools to illustrate probability concepts. For example, use GeoGebra software to perform a dice rolling experiment.
2.3. Applying GeoGebra software in teaching Statistics and Probability
GeoGebra is a dynamic mathematics software that can be used in schools, classrooms and any educational environment, with the main purpose of helping to improve students' knowledge of mathematical concepts and processes. GeoGebra was developed by Markus Hohenwarter in 2001 as a master's thesis and allows the use of geometry, algebra and analysis. GeoGebra also has great statistical tools from creating
statistical charts to analyzing data without requiring programming or coding skills.
GeoGebra is of particular interest to educators with limited budgets because it is an open source software package, available for free on the internet. That's why Geogebra software is very popular in Vietnam. In the article "Some measures for teaching the topic Statistics and Probability for 6th grade students with the support of information technology", authors Nguyen Thi Thu Trang and Quach Thi Sen affirm that it is completely possible to apply Use information technology, especially GeoGebra software, when teaching Statistics and Probability. Absorbing and developing that achievement, we researched and found that teachers can integrate GeoGebra software into teaching Statistics and Probability content in grade 10. In this study, GeoGebra software is used to Form conceptual knowledge of Statistics and probability for students based on the need to integrate technology in teaching and learning Mathematics. A model is described as a surrogate through which we can manipulate and explore the properties of an object without the real thing.
2.4. Illustrating the use of GeoGebra software in teaching some concepts of Statistics and Probability in Grade 10
GeoGebra software can support teaching mathematical concepts visually and interactively in the following ways:
Approaching concepts: Teachers use GeoGebra to create objects and then modify them for students to observe. Teachers provide opportunities for students to engage in activities such as analysis, comparison, and synthesis to discover common characteristics of the objects being examined. From there, students recognize the distinctive features of the concept.
Identifying concepts: Using GeoGebra to measure, calculate, and check the attributes of concepts helps students determine whether an object meets the criteria of an idea.
Systematizing concepts: GeoGebra can assist in systematizing concepts, helping students see the relationships between different concepts.
Inductive Teaching Approach to Concept Instruction is a teaching method in which students are guided from specific phenomena and events to general concepts and principles. This method emphasizes students discovering and identifying rules and principles on their own through observation, practice, and drawing conclusions. To illustrate the use of GeoGebra software in teaching some concepts through the inductive approach, we propose the following steps:
Step 1: Designing Models and Creating Problems
The teacher uses GeoGebra to design models and create problems that help students easily visualize and observe the object of study. Step 2: Teaching Process
Initiation: Let students observe the example, from there the teacher leads students into the definition.
Concept Formation: The teacher uses GeoGebra to pose related questions and asks students to answer them. This process allows students to discover the common attributes necessary to form the concept. Once students identify these common attributes, the teacher introduces the name of the concept and asks students to state the definition in a general context.
Consolidation: The teacher presents a problem that utilizes GeoGebra to reinforce the concept.
In this article, we used GeoGebra software to teach the concept of "Arithmetic mean" in Lesson 2 -Characteristic numbers measuring central tendency for the data sample are not grouped and to state the formula for calculating the probability of an event in Lesson 5 - Probability of Events (Mathematics 10, Canh Dieu textbook, Volume 2).
2.4.1. Teaching the Concept of "Probability of an Event"
In this teaching scenario, the teacher uses GeoGebra software to build a model that observes the results after multiple rolls of 2 dice. From this, the teacher and students can draw conclusions and state the formula for calculating the probability of an event:
n(A)
The probability of event A, denoted as P(A), is the ratio where n(A) and n(Q) are the numbers
respectively of elements in sets A and Q. Thus, P(A)=
n(A) n(a).
2.4.1.1. Design the dice model on GeoGebra software
Step 1: Access the link: https://www.geogebra.org/ Step 2: Create a multi-sided model of two dice
(Photo of dice model: https://vi.pngtree.com/freepng/dice-faces-two-photo_14220242.html ) Step 3: Name the two dice.
Step 4: Enter the command values kqa, kqb, Sum (Total number of dots of 2 dice after rolling)
/» 0 i =N 4- Ш Л С:
Button Image List KQ = ( ) : н
= «J. {}} kqa= SetValue(kqa,Append(kqa,a))
Sum = o j 'kqb= :SetValue(kqb,Append(kqb,b))
kpa = {} • ^r kqb = ft • / Sum =SetValue(Sum,Append(Sum,c))
Step 5: Append the sow button and the sow button again from the beginning
Step 6: Enter the command to calculate "Frequency of appearance of total dots".
2.4.1.2. Teaching process
Teacher's activities
Student's activities
• Evoke motivation:
The teacher accesses the pre-created QR code and presses the F9 key to roll two dice on Excel software. The teacher asks students to observe the results after sowing many times.
Question: Can you predict the outcome of the game beforehand?
- The teacher comments on the students' answers. From there, the teacher guides the students to define "Random experiment and sample space". • Concept formation: The teacher provides the definition:
- Students observe the results and think about answering the question.
- Students cannot predict the outcome before rolling the dice.
"There are experiments where we cannot predict the outcome in advance, even though we know the set of all possible outcomes. Such experiments are called random experiments."
Activity 1:
The teacher has students observe an experiment of rolling two dice 5 times using the GeoGebra software. Students are required to fill in the results in Table 1 and determine the set Q
Table 1
Dice 1 Dice 2
Roll 1
Roll 2
Roll 3
Roll 4
Roll 5
The results after 5 ro ls by the teacher on the GeoGebra software.
The teacher comments on the students' answers and draws a conclusion.
Definition:
There are experiments where we cannot predict the outcome in advance, even though we know the set of all possible outcomes. Such experiments are called random experiments.
The set Q known as the sample space of the experiment. Activity 2:
The teacher presents a slide, describing the scenario using GeoGebra software. Students are asked to observe and answer questions.
Write the set of outcomes corresponding to each event below:
+ The sum of the dots on two dice is 5. + The sum of the dots on two dice is 7. + The sum of the dots on two dice is not 5.
- From the student's answer, the teacher asks a student to articulate the concept.
Table 1
Dice 1 Dice 2
Roll 1 5 1
Roll 2 2 3
Roll 3 6 4
Roll 4 5 4
Roll 5 3 2
О = {(5, 1), (2, 3), (6, 4), (5, 4), (3, 2)}
Let A be "The sum of the dots on two dice is 5".
Let B be "The sum of the dots on two dice is 7".
Let C be "The sum of the dots on two
dice is not 5".
Then:
A= {(2, 3), (3, 2)}
B= 0
C= {(5,1), (6,4), (5,4)}
A student states the concept of an
■ The teacher then finalizes the definition of an event.
Activity 3:
Group discussion and completion of learning tasks The teacher presents a slide, describing the situation using GeoGebra software. Students are asked to observe and answer questions. Learning tasks:
Observe the frequency of occurrence "Sum of the dots on two dice".
a) Define event A = "The sum of the dots on two dice is 5" and calculate the number of elements in the sample space.
n(A)
b) Calculate the ratio
n(n)
- From the students' answers, the teacher summarizes the knowledge:
The probability of event A, denoted as P(A), is the ratio
n(A)
-, where n(A) and n(Q) are the numbers respectively of
n(nj
elements in sets A andQ. Thus, P(A)=
n(A) n(a).
• Consolidate concepts
The teacher assigns the following example to the students: The teacher rolls two fair dice on the GeoGebra software. Exactly one die shows a face with 1 dot.
a. Describe the sample space.
b. Define the following events:
A: "Exactly one die shows a face with 1 dot." B: "The total number of dots on the two dice is less than or equal to 5, and exactly one die shows a face with 1 dot." Calculate P(A) and P(B).
event:
- A random event (event) is a subset of the sample space Q
- The set 0 is an impossible event.
- The space Q is a certain event.
- The complement of event A, denoted
as A and A = Q \ A
Students observe the results on the
GeoGebra software and answer the
questions.
a) n (Q ) = 6.6=36
A= {(2, 3), (3, 2)} => n(A)=2.
n(A) _ 2
n(n) = 36
b)
Students work individually to complete the example. Then, one student goes to the board to solve it, while the other students observe, compare with their own work, and provide comments. Student's solution: a. Sample space: Q = {(1, 1); (1, 2); (1, 3); (1, 4); (1, 5);
(1, 6); (2, 1) (2, 6); (3, 1) (3, 6); (4, 1) (4, 6); (5, 1) (5, 6); (6, 1); (6, 6)}. ^ n(Q)=36.
(2, 2) (3, 2) (4, 2) (5, 2)
(2, 3); (2, 4); (2, 5) (3, 3); (3, 4); (3, 5) (4, 3); (4, 4); (4, 5) (5, 3); (5, 4); (5, 5)
(6, 2); (6, 3); (6, 4); (6, 5)
b. Define the following events:
+ A = {(1, 1); (1, 2); (1, 3); (1, 4);
(1, 5); (1, 6); (2, 1); (3,1); (4,1); (5,1);
(6,1)}
* n(A) = 11=>P(A)= = 11 v ' v ' n(n) 36
+ B = {(1, 1); (1, 2); (1, 3); (1, 4); (2, 1);
(3,1); (4,1)}
=> n(B)=7
=> P(A)= -H = —. n(n) 36
2.4.2. Teaching the concept of "The arithmetic mean".
In this teaching situation, teachers use GeoGebra software to develop statistical data problems, students observe and create frequency tables. From there, we can derive the concept and formula for calculating the mean:
The mean of a sample of n statistical data is equal to the sum of the data divided by the number of
those data. The mean of the data samples x1, x2, ..., Xn is equal to
_ X-1 + X2 + — + xn X = -
n
2.4.2.1. Create math problem on GeoGebra software.
Step 1: Create a data table with five rows and six columns.
Step 2: Set up a random number sequence of 30 numbers. We choose the smallest value (min) to be 20 and the largest value (max) to be 30.
О ■ s
о
zN
» s © 10 ©
i3 Q> =
4 Л C : *
N = a b
= 30
II = Sequence( Random Between (20,30), ¡,1,30,1)
= {30. 25, 23. 29, 23. 23. 27. 28. 29, 26, 20, 26. 27, 23. 22, 23. 23, 27. 28, 26. 22. 26, 24. 21, 20, 23. 23. 21. 20. 27}
Input Sequence(RandomBetween(20(30)li(1,30,1). Then press enter, 11 appears with a random number sequence in the range (20;30).
Step 3: Create a data table from the random number sequence just created in step 2. First, we create a matrix of six columns and five rows from the random number sequence just created in the step above.
Next, we create a data table.
Step 4: Create a value table and frequency table. First, we create a table of values.
Next, we create frequency table.
Step 5: Display statistical results on GeoGebra software.
M I
"Jj One Variable Analysis
Two Viable Regression Analysis jp^jj Multiplewariable Analysis
Step 3: Choose One variable analysis tool
/ m Л CE, »
21 22
25
26
Value Frequency
Q> =
I
Step 2: Enter data from the value table and corresponding frequency table in two columns. After that, use the mouse to
highlight the value and frequency table as shown.
2.4.2.2. Teaching process
Teacher's activities
Student's activities
• Evoke motivation:
- Teacher shows the topic on GeoGebra software: A store selling construction materials lists the cement bags sold in 30 days as shown in the following table:
30 25 23 29 23 23
27 28 29 26 20 26
27 23 22 23 23 27
28 26 22 26 24 21
20 23 23 21 20 27
- Teacher asks students to use CASIO calculator to calculate the mean of 30 numbers from the data table above.
- Teacher gives results on GeoGebra and comments on students' answers.
Statistics
n 30
Mean 24.5
a 2.8607
s 2.9096
Ix 735
Ix2 18253
Min 20
G1 23
Median 23.5
Q3 27
Max 30
- Students think and give answers.
- Students report the results:
The total number of bags of cement sold in 30 days is 735 (bags). The arithmetic mean is 735:30 = 24.5.
From there, the teacher leads students into the definition and formula for calculating "The mean of a statistical data sample". • Concept formation:
The teacher introduces the concept: The mean of a sample of n statistical data is equal to the sum of the data divided by the
number of those data. The arithmetic mean of data samples xi, x2, ..., Xn is equal to
x = ■
X1 + X2 + ••• + xn
n
Activity 1: Teacher recalls the concept of frequency and how to create a frequency distribution table.
- The frequency of a value x is the number of times that value x appears in a statistical data table.
- How to create a frequency table:
Step 1: Determine the values xi,x2,...,xk in the given series of n statistical data (k < n) and determine the frequenci es ni,n2,...,nk of these values.
Step 2: Gather the results found in the previous step (values Xi,
frequency ni) into a table. In the table, xi values are usually
arranged in ascending or descending order.
Activity 2: Teacher lets students observe the problem on
GeoGebra software. Ask students to discuss in pairs and
complete the learning task.
Learning tasks:
From the cement bag statistics table of a store, let's:
a) Make a frequency distribution table
b) Calculate the mean of the above data sample.
- Teacher asks representatives of some groups to answer and other groups to comment.
- Teacher shows accurate results on GeoGebra software. a)
b)
- Students do the exercise and answer the questions.
Value Frequency
20 3
21 2
22 2
23 8
24 1
25 1
26 4
27 4
28 2
29 2
30 1
N = 30
b)
20.3 + 21.2 + 22.2 + —+ 30.1
x =-—-= 24.5
30
- Students observe the results on Geogebra software.
- From the students' answers, the teacher summarizes the knowledge:
The arithmetic mean of the statistical data sample in the frequency distribution table is:
n1x1 + n2x2 +----+ nkXk
x =-
n1+n2 +-----+nk
where: nk is the frequency of the value xk ,
n = ni + n2 +...+ nk.
• Consolidate concepts
- The teacher gives students the following example:
The 50m running time of 20 students is recorded in the table below:
Time (seconds) 8,3 8,4 8,5 8,7 8,8
Frequency 2 3 9 5 1
The mean running time of students is: A. 8,54 B. 4 C. 8,50 D. 8,53
- Teacher asks students to go to the board to do the assignment.
- Teacher reviews students' work.
- Teacher guides students to use GeoGebra software to compare the results of the above exercise.
(Based on Step 5, section 2.4.2.1. Create problem on GeoGebra software).
- Students work individually to complete the example.
The answer:
n1 + n2 + n3 + n4 + ns = n = 20 The mean running time of students is:
n1x1 + n2x2 + n3x3 + n4x4 + nsxs
X =-
n1+ n2+n3+n4 + ns _ 2.8,3 + 3.8,4 + 9.8.5 + 5.8,7 + 1.8,8
= 20 = 8,53
Choose D.
- Students in the class observe your work, compare with your homework and make comments.
- Students observe the results on Geogebra software.
- Students listen to the instructions of the teacher and re-perform the operations on the Geogebra software applied to the exercise.
The application of GeoGebra software in teaching concepts on the topic of Statistics and Probability in grade 10 has contributed to increasing interactivity and active participation and excitement of students. Thereby improving problem solving capacity and learning outcomes of students on this topic.
3. Conclusion
Teaching the topic of Statistics and Probability in grade 10 with the support of information technology helps ensure visualization and mathematical modeling; Promote students' positivity and thinking in the process of discovering new knowledge. The research results of the article expand opportunities for using GeoGebra in teaching and support teachers in teaching some concepts of Statistics and probability for grade
10. However, for teaching and application Information technology is highly effective, teachers need to carefully research lessons, carefully prepare lesson plans, be proficient in GeoGebra software and plan carefully and meticulously. To achieve high results, teachers need to flexibly apply the proposed measures combined with available experience in the process of building lessons suitable for each student. References
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©Pham Minh Chau, Truong Thi Khanh Chi, Le Thi Thoa, 2024
