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ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIV, Issue III, March 2025
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Students’ Errors in Solving Word Problems Involving Angles of
Elevation and Depression and Performance Level in Grade 9
Trigonometry: Basis for Development of a Teacher-Made Material
Reñel M. Sabanal and Analyn A. Guevara
Saint Mary’s University
DOI : https://doi.org/10.51583/IJLTEMAS.2025.140300026
Received: 17 March 2025; Accepted: 29 March 2025; Published: 08 April 2025
Abstract: This study aimed to identify and classify the errors committed by Grade 10 students of a public high school and private
high school when solving word problems involving angles of elevation and depression and to determine their performance level
in trigonometry. Employing sequential-explanatory and correlational designs along with qualitative and quantitative approaches,
the study used Newman’s Error Hierarchical Model to analyze the errors committed by the students which included reading
errors, comprehension errors, transformation errors, process skills errors, and encoding errors. Data were gathered using
worksheet, summative test and interview. Results showed that students’ reading errors were very low but very high in terms of
comprehension error, transformation error, process skills error and encoding error. The summative test in trigonometry revealed
that majority of the students performed poorly, with only a small percentage achieving fairly satisfactory or satisfactory results.
Moreover, the findings revealed that there is a significant relationship between the errors committed by the students and their
performance score in trigonometry. Based on the findings, a teacher-made material with corrective instructional activities which
may be adopted by mathematics teachers in addressing errors in solving word problems in trigonometry was developed.
Keywords: Comprehension error, encoding error, Newman’s error hierarchical model, process skills error, transformation error
I. Introduction
Mathematics is a vital medium for real-world application. Learning mathematics teaches us not just how to count but also how to
enhance our logical, analytical, and systematic thinking skills and use them in everyday life (Sevgi & Arslan, 2020). Learners
continue to study mathematics throughout college, with many specializing in statistics, applied mathematics, or theoretical
mathematics.
The Philippine Mathematics Curriculum in the current K to 12 curriculum provides that different branches of mathematics are
effectively delivered to students quarterly (Balagtas et al., 2019). One of these branches is trigonometry. Trigonometry covers the
study of triangles, the relationships between their sides and angles, the functions of sine and cosine, tangent and cotangent, and
secant and cosecant (Walsh et al., 2017). Trigonometry is a critical topic for students because it enables them to be prepared for
higher mathematics like calculus (Hidayati, 2020). Its application extends to professional fields like architecture, engineering,
cartography, and other advanced fields (Galarza, 2017).
The Third International mathematics and Science Study (TIMSS) 2019 data revealed that Filipino students’ poor mathematical
performance has placed the country in the 57th rank out of 58 countries worldwide. Another report revealed that students from
Cagayan Valley in the Philippines, Region 2, perform poorly in mathematics where the average grade for Grade 6 pupils, Grade
10 and Grade 12 students are respectively 36.66%, 36.91%, and 31.02% based on the National Achievement Test (2019).
One method that can be utilized to evaluate and examine students’ errors is Newman’s Error Hierarchical Model which originated
from a 1970s investigation of mathematical language difficulties which has 5 phases namely reading, comprehension,
transformation, process skills, and encoding (Lestari et al., 2018). Reading errors stem from inadequate basic reading skills.
Comprehension errors arise from misunderstanding the problem's requirements. Transformation errors occur when students
struggle to translate problem information into mathematical models or select suitable problem-solving methods. Process skill
errors involve incorrect procedures, leading to procedural, arithmetic, or incomplete solutions. Encoding errors entail correct
problem solving but inaccurate or omitted answer presentation.
As of November 2016, information from the Department of Education (DepEd) revealed that the organization lacked 235 million
instructional and other learning materials (Umil, 2017) putting educational process in danger. In particular, Maligalig et al. (2011)
found that one prevalent concern in the country's current educational system is the lack of learning materials and necessary
teaching resources in the mathematics classroom.
In this study, Newman’s Error Hierarchical Model was utilized to determine the errors committed by students, the results of which
was used to craft a teacher-made material that can efficiently help teachers handle student errors and effectively help students
become better at solving word problems involving angles of elevation and depression. This study identifies common errors in
math word problem solving, benefiting DepEd, administrators, teachers, students, and researchers.
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Theoretical Framework
Newman's Error Hierarchical Model, integrated with a constructivist teaching approach based on Piaget's theory, offers insight
into handling student errors and learning new concepts. This framework prioritizes error analysis to understand reasoning behind
wrong responses, aiding teachers in identifying areas needing support. Learning is viewed as collaborative, fostering debate and
teamwork through error analysis, promoting deeper understanding among students.
Conceptual and Analytical Framework
This study aimed to identify and classify students’ committed errors in solving word problems involving angles of elevation and
depression based on Newman’s Error Hierarchical Model and to investigate their performance level in trigonometry.
Newman’s Error Hierarchical Model has been shown to be a reliable model for error analysis and is used by mathematics teacher
researchers. As indicated in the study of Nurmeidina and Rafidiyah (2019), trigonometry is a very important subject in
mathematics. However, it is alarming to note that many students are underachievers in trigonometry. Figure 2 displays the
research paradigm of the study.
Figure 1. Research Paradigm
The study examines students' errors in trigonometry word problems on angles of elevation and depression. Figure 1 illustrates the
relationship between these errors and trigonometry performance. Newman’s Error Hierarchical Model guides error analysis,
supplemented by triangulated data from worksheets and interviews to develop targeted teaching materials addressing common
errors.
Statement of the Objectives
The study aimed to analyze the errors that Grade 10 students at Cagasat High School and Saint John Berchmans High School
Incorporated for the SY 2023-2024 commit in solving word problems involving to angles of elevation and depression.
Specifically, the objectives of the study were the following:
1. Identify and classify the common errors of students in solving word problems involving angles of elevation and
depression using Newman’s Error Hierarchical Model;
2. Determine the level of performance of students in trigonometry for Grade 10;
3. Describe the relationship between the number of errors committed and performance of Grade 10 students; and
4. Develop teacher-made material based on the identified errors of the students in solving word problems involving angles
of elevation and depression and their performance in trigonometry.
Statement of Null Hypothesis
1. There is no significant relationship between the errors committed and the performance of Grade 10 students.
II. Methodology
This study utilized sequential-explanatory and correlational designs along with qualitative and quantitative approaches.
Quantitative analysis was employed to calculate the number of errors committed by the students per error indicator and to
determine their performance level in trigonometry. The qualitative analysis was carried out by describing each student's errors
when solving word problems based on Newman's Hierarchical Error Model. Moreover, it was used to identify the reason behind
the errors committed by the students through the use of interviews. Lastly, a correlational study was done to examine the
relationship between the errors made by the students when solving word problems and their performance score in trigonometry.
1
In the course of analysis, the study will explore potential relationships 2
between the committed errors of the students when solving word problems and 3
their performance in trigonometry. Figure 2 displays the research paradigm of the 4
study. 5
6
7
8
9
10
DEVELOPMENT OF A
TEACHER-MADE
MATERIAL
Errors Committed by
Students
Reading Error
Comprehension Error
Transformation Error
Process Skills Error
Encoding Error
Students’ Performance in
Trigonometry
1. illustrates the six trigonometric ratios:
sine, cosine, tangent, secant, cosecant,
and cotangent.
2. finds the trigonometric ratios of
special angles
3. illustrates angles of elevation and
angles of depression
4. uses trigonometric ratios to solve real-
life problems involving right triangles.
5. illustrates the laws of sines and
cosines.
6. solves problems involving oblique
triangles.
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The researcher has selected secondary schools in Cordon, Isabela as the research environment of this study. The town of Cordon,
located in Isabela which is in Cagayan Valley, a part of Luzon, has a limited number of secondary schools. Specifically, Cordon
comprises five public secondary schools and two private secondary schools. Because of the relatively low count of both public
and private secondary schools, the researcher selected one representative for public secondary school and one representative for
private secondary school.
Table 1. List of Selected Schools in Cordon, Isabela
Classification
Chosen School
Public
Cagasat High School
Private
Saint John Berchmans High School Incorporated
Moreover, Table 1 provides the location of the two selected schools in Cordon, Isabela. Saint John Berchmans High School
Incorporated (SJBHSI), a private institution, is located along National Highway in Brgy. Magsaysay, Cordon, Isabela and founded
in 1962. In contrast, Cagasat High School (CHS), a pioneer public secondary school located on Tasani St. in Brgy. Gayong,
Cordon, Isabela, was established in 1971.
The respondents of the study were the Grade 10 students who studied trigonometry in their ninth grade enrolled during the school
year 2022-2023 in the selected schools in Cordon, Isabela. Through the use of Slovin’s Formula, 97 Grade 10 students at Cagasat
High School and 128 Grade 10 students at Saint John Berchmans High School Incorporated students were selected with a 0.05
margin of error. Slovin's Formula was applied to determine the appropriate sample size from the stated population of the study.
Table 2 shows the number of Grade 10 students according to their sex and age. The table displays that there are 110 male students
(48.89%) and 115 female students (51.11%) selected in this study as respondents. There are more female respondents than male
respondents in this study. The table further displays that there are 111 respondents (49.33%) falling under the age of 14 years old,
110 respondents (48.89%) fall under the age of 15 years old, and four respondents (1.78%) fall under the age bracket 16-19 years
old.
The study included a total of 225 respondents, drawn from the nine sections of Grade 10 students at Cagasat High School and
Saint John Berchmans High School Incorporated. The selection process employed systematic random sampling, specifically
choosing every third student present both during the administration of the worksheet and summative test in each class. To ensure
confidentiality, each respondent involved in the study was assigned a pseudonym. This measure was taken to protect the privacy
of the participants and maintain the ethical standards of research. In the data collection process, all selected participants were
interviewed and asked to read the word problems presented on the worksheet. This interactive approach allowed the researcher to
directly assess each student's reading ability. For those students demonstrating pattern errors, follow-up questions were
administered to gain insights into their thought processes and reasoning.
Table 2. Demographic Profile of Grade 10 Respondents
Name of Schools
Total
Percent
CHS
SJBHSI
Age
14 years old
80
31
111
49.33
15 years old
15
95
110
48.89
16- 19 years old and above
2
2
4
1.78
Sex
Male
54
56
110
48.89
Female
43
72
115
51.11
The researcher employed three research instruments namely worksheets on word problems involving angles of elevation and
depression, interview guidelines, and a summative test on trigonometry. The three research instruments were validated by three
validators with the use of a validation checklist. The researcher utilized five items of word problems involving angles of elevation
and depression to evaluate the students' errors in trigonometry. Moreover, the researcher made a 50-item summative test in
trigonometry which was used to measure the performance level of students in trigonometry. Lastly, the researcher used semi-
structured interviews as a tool to discover the causes of students' committing such errors from the perspective of the students
themselves.
The researcher carried out a quantitative and qualitative content analysis on the collected data from the prepared worksheet.
Newman’s Error Hierarchical Model was utilized to conduct data analysis in identifying the actual errors made by the students
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when solving the word problems involving angles of elevation and depression. The errors committed by the students were coded
in each domain of the Newman Error Hierarchical Model from each word problem. Furthermore, it was acknowledged that
students can commit multiple errors in their submitted worksheets. The interview provided a deeper insight into the students’
errors from their perspective. Students' performance in trigonometry was described in frequency, percent and averages, maximum
value, minimum value, and standard deviation.
Inferential data analysis was conducted using Statistical Product and Service Solutions (SPSS). Due to the non-normality of the
data, Spearman’s Rho Correlation Test was selected as the appropriate statistical tool. To investigate these relationships, the study
specifically focused on relating the number of errors per error indicator committed by the students with their performance scores
in trigonometry.
In the process of developing teacher-made material, the focus was directed toward the learning competency associated with the
application involving right triangles. The teacher-made material includes corrective instruction activities, strategically
recommended to rectify the errors observed in the students' problem-solving skills. Within the teacher-made material, details are
provided outlining the types of errors commonly observed, proposing corrective instruction activities, and describing the
procedures for implementing these activities.
The researcher conducted the study not for personal gain, but to improve mathematics instruction and learning. There was no
conflict of interest in the conduct of the study. The researcher ensured that the instruments that were ready for usage, coding, and
analysis before the actual study. The profiles of the respondents and the data collected were treated with utmost confidentiality.
The collected tests and worksheets were stored in a secure location. Specifically, the answered worksheets and tests were placed
in expanding envelopes. The study's vulnerable populations (those under the age of 18) were not exploited and the researcher
considered their needs as well as any difficulties that might arise in their capacity to provide complete informed consent. The
involvement of the respondents in the study carried some risks. The respondents may be inconvenienced or at risk due to the time
taken away from them. The existence of an informed consent form in this study served as a proactive measure to address and
mitigate potential risks.
III. Results and Discussion
Section 1. Common Errors of Grade 10 Students in Solving Word Problems Involving Angles of Elevation and Depression
Table 3 provides a detailed breakdown of student errors organized by specific indicators according to Newman's Error
Hierarchical Model. Each error is thoroughly examined and discussed in succeeding subsections for clarity.
Table 3. Summary of Error Codes
Error Codes
Item 1
Item 2
Item 3
Item 4
Item 5
(%)
n
%
n
%
n
%
n
%
n
%
R1
30
13.3
23
10.2
2
0.9
24
10.7
1
0.4
35.5
R2
9
4.0
1
0.4
43
19.1
5
2.2
29
12.9
7.72
C1
15
6.7
45
20.0
62
27.6
52
23.1
101
44.9
24.6
C2
121
53.8
165
73.3
134
59.6
167
74.2
105
46.7
61.53
T1
47
20.9
88
39.1
118
52.4
103
45.8
158
70.2
45.68
T2
88
39.1
121
53.8
86
38.2
117
52.0
55
24.4
41.5
P1
51
22.7
92
40.9
131
58.2
105
46.7
163
72.4
48.18
P2
113
50.2
123
54.7
76
33.8
120
53.3
56
24.9
43.38
E1
84
37.3
125
55.6
152
67.6
134
59.6
173
76.9
59.4
E2
94
41.8
94
41.8
57
25.3
91
40.4
48
21.3
34.12
Reading Errors
Failure to read symbols or signs that prevent students from continuing the calculation process is called a reading error. There are
reading errors in the different word problems committed by the students. Specifically, 39 students (17.3%) encountered reading
errors with the first word problem. The second word problem posed reading error challenges for 24 students (10.7%). The third
word problem presented reading errors for 45 students (20%). Moving on to the fourth word problem, 29 students (12.9%)
struggled with reading errors. Lastly, the fifth word problem perplexed 30 students (13.3%) who struggled with reading. On
average, the reading error rate across these word problems is at 14.84%. The types of students’ reading error in this domain were
divided into two, namely: (R1) the student cannot find the meaning of the mathematical symbols, and (R2) the student cannot
read the questions correctly. R1 indicates that students struggle to understand the interpretation of mathematical symbols
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contained in the word problem while R2 implies that the student is having difficulty in reading the words contained in the word
problems.
Researcher: Can you read the word problem #1?
Axel: (Reads the 1
st
word problem but he read the symbol as percent)
Researcher: What do you call this symbol again? (points at symbol)
Axel: Percent Sir.
(The researcher explains that the symbol should be read as degree/s)
Researcher: Can you read the word problem #3?
Belle: (Reads the 1
st
word problem but he read the word “leans” as lens)
Researcher: Can you read the first statement of word problem #3 again?
Belle: (Reads the 1
st
statement and he read again the word “leans” as lens)
(The researcher explains that the pronunciation of the word “leans” is /lins/)
The majority of students demonstrated proficiency in overcoming reading errors when reading mathematical symbols and words
contained in the word problems involving angles of elevation and depression. However, a small number of students, including
Axel and Belle, encountered difficulties in reading the problems. Axel acknowledged misinterpreting the mathematical symbol,
reading the ° as a percent due to his prior knowledge. Some students chose to skip reading the ° symbol altogether, while others
mistakenly identified it as a degree Celsius symbol. Approximately 35.5% of the respondents exhibited an indicator of reading
error R1.
Meanwhile, R2 exemplifies specific instances of misreading like "leans" being read as "lens" in the 3rd word problem, "attached"
as "attacked" in the 1st word problem, "lighthouse" as "lithows" in the 4th word problem, and "diver" as "driver" in the 5th word
problem. However, only 7.72% of respondents displayed the R2 reading error indicator.
Among the reading errors observed in students, it was noted that they often bypass unfamiliar words, leading to potential
misunderstandings. Additionally, some students tended to incorrectly articulate mathematical symbols, which could potentially
impact the accuracy of their responses. As Lubis et al. (2021) have found, students face challenges in identifying the given
information within the word problem which hinders their ability to solve the word problem.
Comprehension Errors
Comprehension errors refer to situations where students can read the questions, yet struggle to grasp the underlying meaning,
resulting in their responses not aligning with the intended inquiry (Lestari et al., 2018). Comprehension errors were experienced
by 136 students in the 1st word problem (60.4%), 210 students in the 2nd word problem (93.3%), 196 students in the 3rd word
problem (87.1%), 219 students in the 4th word problem (97.3%), and 206 students in the 5th word problem (91.6%) with an
average of 85.94%. In this study, comprehension error arises when students cannot illustrate the given problem (C1) and when
students do not understand the problem (C2).
Figure 2. Clara’s Answer to the Fourth Word Problem with C1 Error
Figure 2 shows Clara's solution to the fourth word problem. Clara appears to be experiencing a comprehension error in indicator
C1, which is shown by a yellow box. Clara supplied a solution, but she did not provide an illustration. The lack of an illustration
may limit a comprehensive grasp of her answer, particularly in the step of determining the appropriate trigonometric ratio. This
error may result in an incorrect answer and conclusion.
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Figure 3. David’s Answer to the Third Word Problem with C2 Error
Figure 3 displays David's response to the third word problem. It appears that David is encountering a comprehension error in
indicator C2, which is highlighted with a yellow box. David misplaced one of the provided pieces of information, leading to an
incorrect illustration. David mistakenly recorded "4" as the height of the wall, when it should have been the length of the ladder.
This misstep led to an inaccurate illustration, directly impacting the trigonometric ratio. Consequently, this error in the
trigonometric ratio can lead to an incorrect answer and an inaccurate conclusion.
Some students also have difficulty representing angles of depression in their illustrations, incorrect placement of the provided
angle, and misplacing an object to the triangle. Overall, comprehension errors were experienced by students while solving word
problems, with a percentage of 24.6% and 61.53% for C1 and C2, respectively. These errors can potentially lead to a
misinterpretation or misunderstanding of the solution. Visual representations often play a crucial role in conveying complex
concepts accurately. As such, encouraging the students to include an illustration alongside their solution would greatly contribute
to ensuring the correctness and clarity of their answers (Prasetyaningrum et al., 2022).
Transformation Errors
Transformation errors result from a lack of familiarity with the appropriate formula to employ in problem-solving (Oktafia et al.,
2020). Among the respondents, 135 students (60%) encountered transformation errors in the 1st word problem. In the 2nd word
problem, 209 students (92.9%) faced similar challenges. This trend continued with 204 students (90.7%) experiencing
transformation difficulties in the 3rd word problem, 220 students (97.8%) in the 4th word problem, and 213 students (94.7%) in
the 5th word problem. On average, this results in an overall transformation rate of 87.22%. In this study, transformation errors
particularly refer to instances where students struggle with selecting the correct mathematical model (T1) and the failure to write
down the mathematical model completely and accurately (T2).
Figure 4. Hector’s Answer to the Third Word Problem with T1 Error
Figure 4 illustrates Hector's response to the third word problem. It indicates that Hector encountered a transformation error in
indicator T1, marked by a yellow box. While Hector can comprehend and illustrate the problem, he faces difficulty in selecting
the appropriate trigonometric ratios for the given scenario. This error in selecting the appropriate trigonometric ratio led to the
absence of an answer and a lack of a conclusion. This indicates a specific area of challenge for Hector in applying trigonometric
concepts to problem-solving.
Figure 5. Ian’s Answer to the First Word Problem with T2 Error
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Figure 5 showcases Ian's response to the first word problem. Ian faced a transformation error in indicator T2, marked by a yellow
box. Despite providing an accurate illustration of the word problem, Ian made an error in writing the appropriate trigonometric
ratio. He wrote the trigonometric ratio tangent instead of sine when attempting to express the trigonometric ratio. Ian explained
that he selected and wrote tangent because the adjacent side of the reference angle was not mentioned in the word problem.
Students with an error of T1 experiencing comprehension errors while solving word problems were at 45.68%, whereas in T2,
this percentage was lower at 41.5%. Understanding and correctly applying mathematical equations is crucial in various areas of
mathematics, especially in geometry, trigonometry, algebra, and calculus (Star et al., 2015). Encountering transformation errors in
this context indicates that the students may need further practice and guidance in this area. In many cases too, students
demonstrate an understanding of the problem statement, but struggle with determining the correct sequence of steps required to
arrive at the solution accurately (Yuliana et al., 2021). This could involve revisiting the definitions of trigonometric ratios and
practicing their application in different types of problems.
Process Skills Errors
Process skills errors were experienced by 164 students in the 1st word problem (72.9%), 215 students in the 2nd word problem
(95.6%), 207 students in the 3rd word problem (92%), 225 students in the 4th word problem (100%), and 219 students in the 5th
word problem (98.2%) with an average of 91.56%. This shows that process skills errors are quite high. In this study, process skills
errors mean that students have no solution (P1) and student does not correctly apply the steps in detail and systematically (P2).
Figure 6. Madison’s Answer to the Fourth Word Problem with P1 Error
Figure 6 presents Madison's response to the fourth word problem. Madison encountered a process skills error in indicator P1,
highlighted by a yellow box. She provided a precise illustration of the given word problem and correctly supplied a trigonometric
ratio to solve for the unknown. However, despite initially offering an accurate illustration and providing an appropriate
trigonometric ratio, she lacked a solution and answer. Additionally, Madison attempted to write the conclusion of the word
problem, but since she did not have an obtained answer, her conclusion was incomplete, also indicating an encoding error.
Figure 7. Nelson’s Answer to the First Word Problem with P2 Error
Figure 7 reveals Nelson's response to the first word problem. Nelson encountered a process skills error in indicator P2, indicated
by a yellow box. Nelson committed a process skill error in providing the correct answer despite providing a precise illustration of
the word problem and using the correct trigonometric ratio. Nelson has exhibited difficulties in using a scientific calculator to
execute his calculations accurately. According to Nelson's interview, he mentioned multiplying 100 and sin 18 instead of
obtaining their quotient.
Siskawati et al. (2021) also found that students exhibit weaknesses in their grasp of process skills, particularly in comprehending
the mathematical equation presented on the word problem. In this study, process skills errors were experienced by students while
solving word problems, with a percentage of 48.18% and 43.38% for P1 and P2, respectively. Students fail to develop systematic
steps in problem-solving in a consistent, detailed, and accurate manner. In relation to this, it is crucial to provide students with
clear instructions and ample practice opportunities (Juneidi et al., 2015).
Encoding Errors
An encoding error arises when a student successfully arrives at a solution but encounters difficulty in articulating the concluding
statement of the answer. Encoding errors were experienced by 178 students in the 1st word problem (79.1%), 219 students in the
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2nd word problem (97.3%), 209 students in the 3rd word problem (92.9%), 225 students in the 4th word problem (100%), and
221 students in the 5th word problem (98.2%) with an average of 93.5%. In this study, encoding errors refer to students not
writing the conclusion (E1) and students not able to answer the problem with the right conclusion (E2).
Figure 8. Pamela’s Answer to the First Word Problem with E1 Error
Figure 8 displays Pamela's response to the first word problem. Pamela encountered an encoding error in indicator E1. She
provided an accurate illustration of the problem and correctly applied the appropriate trigonometric ratio to determine the
unknown, resulting in a precise and correct result despite this encoding error. However, Pamela did not provide a conclusion. By
including a conclusive statement, Pamela could have provided a more holistic and well-rounded response, ensuring that the
recipient of her work gains a thorough understanding of the problem's resolution.
Figure 9. Ralph’s Answer to the Fifth Word Problem with E2 Error
Figure 9 presents Ralph's answer to the fifth word problem. Ralph encountered an encoding error in indicator E2, represented by
the yellow box. Ralph has effectively provided an appropriate illustration of the problem, utilized the correct trigonometric ratios,
and arrived at an accurate solution. However, there was a slight oversight in providing the correct unit of measurement for the
solution. Ralph included feet in his conclusion when the accurate unit of measurement should have been degrees, as the unknown
involved an angle. This emphasizes the significance of careful attention to detail, particularly in specifying units, which is integral
to conveying precise mathematical solutions.
Other students experienced encoding errors, which included the use of improper grammar or an incomplete thought in their
conclusions and the application of incorrect units of measurement. Overall, students encountered encoding errors while solving
word problems, with an average of 59.4% for E1 and 34.12% for E2. These errors may have occurred because of the student's
lack of prioritization, diminished interest, and a certain level of indifference toward formulating an accurate concluding statement.
Section 2. Student’s Performance Level in Trigonometry
Table 4. Mean Percent Scores and Performance Level in Each Learning Competency in Trigonometry
Learning Competencies
Mean Percent Score
Performance Level
1. Illustrates the six trigonometric ratios: sine, cosine,
tangent, secant, cosecant, and cotangent.
42.88
Poor Performance
2. Finds the trigonometric ratios of special angles
32.20
Poor Performance
3. Illustrates angles of elevation and angles of depression
46.92
Poor Performance
4. Uses trigonometric ratios to solve real-life problems
involving right triangles.
34.27
Poor Performance
5. Illustrates the laws of sines and cosines.
19.64
Poor Performance
6. Solves problems involving oblique triangles.
32.91
Poor Performance
Overall
34.80
Poor Performance
holistic and well-rounded response, ensuring that the recipient of her work
gains a thorough understanding of the problem's resolution.
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Table 4 displays the mean percent scores and performance level in each learning competency in trigonometry of Grade 10
students. The analysis of student performance in the first learning competency, reveals a poor performance among the students
with a mean percent score of 42.88%. The table also indicates that students exhibited a poor performance in the second learning
competency with a mean percent score of 32.20%. Table 4 further indicates that students exhibited a poor performance in the third
learning competency as reflected in a mean percent score of 46.92%. Moreover, the table reveals that majority of students
demonstrated a poor performance in the fourth learning competency as evidenced by a mean percent score of 34.27% The table
also indicates that students demonstrated a poor performance in the fifth learning competency with a mean percent score of
19.64%. Lastly, the table reveals that majority of students demonstrated a poor performance in the sixth learning competency
having an average percent score of 32.92%. The overall mean percent score of 34.80% suggests that the students, on average,
performed poorly in trigonometry across various learning competencies. This could indicate a need for improvement in teaching
methods, student engagement, or the overall learning environment.
Section 3. Relationship Between the Committed Errors and the Performance of Grade 10 Students in Trigonometry
Table 5. Correlation of the Number of Committed Errors of the Respondents to their Performance Score in Trigonometry
Error Domains
Error Indicators
Correlation Coefficient
Significance Level
Reading Error
R1
-0.059
0.380
R2
-0.261**
<0.001
Comprehension
Error
C1
-0.258**
<0.001
C2
0.072
0.284
Transformation
Error
T1
-0.355**
<0.001
T2
0.213**
0.001
Process Skills Error
P1
-0.337**
<0.001
P2
0.234**
<0.001
Encoding Error
E1
-0.386**
<0.001
E2
0.308**
<0.001
** Significant at 0.01 level of significance
A study found a significant negative correlation between R1, indicating reading comprehension issues,
and Grade 10 trigonometry performance. More reading errors were linked to poorer trigonometry scores, suggesting reading
difficulties may hinder math performance in this group.
Moreover, the study revealed a significant relationship between comprehension errors on C1, and the performance score of Grade
10 students in trigonometry, . This indicates that fewer student illustrations correlate with
decreased trigonometry scores, suggesting a potential impact on performance.
The study found significant relationships between transformation error indicators, T1 and T2, and trigonometry performance
among students for T1 forT2. T1 errors showed a weak negative correlation , indicating
lower trigonometry scores with more T1 errors. Conversely, T2 errors exhibited a weak positive correlation ,
suggesting slight improvement in trigonometry scores with more T2 errors. This underscores the role of effort and engagement in
learning.
Table 5 reveals significant relationships between process skills errors, P1 and P2, and trigonometry performance ( for
P1, for P2). P1 errors show a weak negative correlation , indicating lower performance scores with
more P1 errors. Conversely, P2 errors display a weak positive correlation , suggesting slight improvement in
performance scores with more P2 errors. This underscores the notion that even imperfect effort and engagement can foster better
understanding and performance.
Furthermore, encoding errors, both E1 and E2, significantly relate to trigonometry performance ( for E1, for
E2). E1 exhibits a weak negative correlation ( ), indicating higher E1 errors coincide with lower trigonometry scores.
Conversely, E2 shows a weak positive correlation ( ), suggesting slight performance score improvement with more E2
errors. In essence, more E1 errors are linked to lower performance scores, while more E2 errors are associated with minor
performance score increases in trigonometry. These errors were statistically significant at the 0.01 level.
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Section 4. Development of a Teacher-made Material Based on the Identified Errors of the Students in Solving Word
Problems Involving Angles of Elevation and Depression and their Performance Level in Trigonometry.
To lessen the committed errors and to enhance students’ performance in the aforementioned learning competency, providing
corrective instruction activities through teacher-made material tailored to students' specific error categories is considered an
effective approach. The researcher developed a teacher-made material that aims to serve as a comprehensive guide for teachers,
enabling the students to improve their performance in learning trigonometry specifically when solving word problems involving
angles of elevation and depression.
These corrective instruction activities are intended to assist students in reducing errors, as outlined in Newman's Error
Hierarchical Model, particularly when solving word problems involving angles of elevation and depression. Furthermore, these
activities aim to foster the development of students' higher-order thinking skills, improved conceptual understanding, enhanced
problem-solving skills, and individualized learning.
Table 6. Corrective Instruction Activities in the Teacher-Made Material
Activity
No.
Activity
Types of Errors
Targeted Learning Competencies
1
Trigonometric Tales:
Heights, Distances, and
Angles!
Reading and
Comprehension
Errors
Illustrates angles of elevation and angles of
depression.
2
SOHCAHTOA
Spectacular: Unraveling
Math Mysteries!
Transformation
Errors
Illustrates the six trigonometric ratios: sine,
cosine, tangent, secant, cosecant, and cotangent
3
Trig-Tacular Calculator
Challenge!
Process Skills
Errors
Uses trigonometric ratios to solve real-life
problems involving right triangles.
Illustrates the six trigonometric ratios: sine,
cosine, tangent, secant, cosecant, and cotangent
4
Enigma Escapade:
Mastering Conclusion
Statements!
Encoding Errors
Uses trigonometric ratios to solve real-life
problems involving right triangles.
Table 6 shows an overview of the corrective instruction activities integrated into the teacher-made material. This instructional
material comprises four distinct activities, each specifically designed with two objectives in mind: firstly, to reduce the occurrence
of the targeted error type committed by the students, and secondly, to enhance their performance in the specified learning
competency in trigonometry. The incorporation of these activities into the curriculum is expected to not only supplement students'
grasp of the subject matter but also contribute significantly to the development of their mathematical proficiency specifically in
solving word problems involving angles of elevation and depression (Bayos, 2020).
In the first activity, titled "Trigonometric Tales: Heights, Distances, and Angles!", students engage in reading and comprehension
exercises to rectify errors associated with understanding word problems. The collaborative nature of the task, where students
work with partners, aims to enhance their ability to read and comprehend trigonometric scenarios. The goal of this activity is to
illustrate angles of elevation and angles of depression. Figure 10 shows some sample items from the first activity.
Figure 10. Sample Items from the First Activity
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Figure 10 displays the first, third, and fifth items from the first activity in the teacher-made material. The items in this activity are
designed to assess and enhance students' reading and comprehension skills related to angles of elevation and angles of depression.
Specifically, the first and second items provide a complete guided statement to assist students in illustrating the given word
problem. The third and fourth word problems include a guided statement that is intentionally left incomplete, challenging students
to illustrate the problem even with partial guidance. The fifth item, without a guided statement, aims to evaluate students'
comprehension by requiring them to independently illustrate the given problem. Additionally, students are required to read the
word problem and the guided question, providing a test of their reading skills. Overall, these activities are crafted to contribute to
the improvement of students' performance in the learning competency of illustrating angles of elevation and angles of depression.
Moving on to the second activity, "SOHCAHTOA Spectacular: Unraveling Math Mysteries!", the focus is on correcting
transformation errors related to providing accurate trigonometric ratios. The learning competency targeted in this activity is the
illustration of the six trigonometric ratios: sine, cosine, tangent, secant, cosecant, and cotangent. Guided questions will assist the
students in overcoming challenges and ensure the correct application of trigonometric concepts. Figure 11 shows some sample
items from the second activity.
Figure 11. Sample Items from the Second Activity
Figure 11 displays the first and third items from the second activity in the teacher-made material. This activity aims to assess
students' ability to transform given word problems and illustrations into trigonometric ratios, with a focus on the trigonometric
ratios: sine, cosine, tangent, secant, cosecant, and cotangent. In the first and second items, a complete question guide is provided
to assist students in identifying the sides of the triangle, determining the unknown, and selecting the appropriate trigonometric
ratio applicable to the problem. The third and fourth word problems also include a question guide intentionally left incomplete,
challenging students to provide a trigonometric ratio even with partial guidance. The fifth item, without a question guide, tests
students' critical thinking skills in determining the appropriate trigonometric ratio for the given problem. This activity is designed
to enhance students' performance in the learning competency of illustrating the six trigonometric ratios, fostering their
understanding of the relationships between angles and sides in a triangle.
The third activity, titled "Trig-Tacular Calculator Challenge!", addresses transformation errors associated with performing
operations in trigonometric equations and utilizing scientific calculators. By requiring the students to execute operations using
calculators, this activity enhances their practical application of trigonometric concepts. The learning competencies addressed
include using trigonometric ratios to solve real-life problems involving right triangles, as well as illustrating the six trigonometric
ratios. Figure 12 shows some sample items from the third activity.
Figure 12. Sample Items from the Third Activity
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Figure 12 features the first and fourth items from the third activity in the teacher-made material. This activity is designed to
evaluate and enhance students' procedural skills in solving for the value of the unknown using a digital scientific calculator. The
focus is on applying trigonometric ratios to solve real-life problems involving right triangles and demonstrating proficiency in
illustrating the six trigonometric ratios. In the first and second items, a complete step-by-step procedure is provided to guide
students in calculating the unknown. The fourth and fifth word problems feature an incomplete procedure, challenging students to
execute the equation and solve for the unknown with partial guidance. The third and sixth items lack a provided procedure,
testing students' ability to independently apply their procedural skills in solving for the unknown using the calculator. This
activity aims to contribute to the improvement of students' performance in the learning competency involving the application of
trigonometric ratios to solve real-life problems associated with right triangles.
Last with the fourth activity, "Enigma Escapade: Mastering Conclusion Statements!", the focus is on rectifying encoding errors
related to writing conclusion statements. The students are encouraged to create conclusive statements based on word problems
and the given answer, addressing challenges in drawing conclusions. The learning competency targeted in this activity is the
application of trigonometric ratios to solve real-life problems involving right triangles. Figure 13 shows some sample items from
the fourth activity.
Figure 13. Sample Items from the Fourth Activity
Figure 13 shows the first and second items from the fourth activity in the teacher-made material. The objective of this activity is
to assess and improve students' ability to articulate conclusions based on the given word problem and its corresponding answer.
The emphasis is on enhancing their performance in the learning competency of applying trigonometric ratios to solve real-life
problems involving right triangles. In the first to fifth items, a conclusion is partially provided with two blanks, requiring students
to fill in the missing information. This challenges them to connect the word problem with the solution and accurately articulate
the appropriate conclusion. The sixth to tenth word problems lack a pre-written conclusion, demanding that students generate a
complete and accurate conclusion based on the given word problem and its solution. This activity aims to further develop
students' proficiency in drawing meaningful conclusions from trigonometric problem-solving scenarios, fostering a deeper
understanding of the application of trigonometric ratios in real-life situations involving right triangles.
Together, these activities form a comprehensive set designed to address specific challenges, and enhance learning competencies in
trigonometry.
IV. Conclusions and Reccommendations
The identified errors, categorized under Newman's Error Hierarchical Model, mainly include reading with most being
comprehension, transformation, process skills, and encoding errors. This highlights the areas that require targeted corrective
instruction activities. The analysis of the students' performance level in the summative test in trigonometry revealed a range of
outcomes, with a proportion falling into the fair to poor performance categories. The correlational analysis established
connections between specific error types in word problems and students' overall performance in trigonometry. Reading errors,
comprehension errors, transformation errors, process skills errors, and encoding errors displayed significant correlations with
students' overall performance in trigonometry. The corrective instruction activities in the teacher-made material are based on the
committed errors made by the students in solving word problems involving angles of elevation and depression and students’
performance in trigonometry. All the error domains are addressed in the material since all of the errors were considered
restrictions.
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The following are the suggestions based on the study's findings and analysis of the data obtained:
1. The researcher recommends that teachers conduct a thorough error analysis to identify the most prevalent types of errors
experienced by students when solving word problems.
2. Given that the most prevalent errors committed by the students were comprehension, transformation, process skills, and
encoding errors, it is recommended that future researchers conduct a similar study across different grade levels.
Additionally, a study focused on evaluating the effectiveness of corrective instruction activities could yield valuable
insights into improving performance in trigonometry. School administrators are also encouraged to implement programs
aimed at enhancing trigonometry instruction across all educational levels from basic education units to higher levels.
3. The researcher also recommends that teachers prioritize familiarizing students with various mathematical word
problems. This approach is essential for enhancing students' problem-solving abilities, ultimately contributing to their
overall proficiency in the subject.
4. To address the underlying causes of errors committed by the students, it is advised that teachers employ suitable
instructional procedures. Additionally, teachers may find it beneficial to integrate the developed teacher-made material
from this study. The utilization of the teacher-made material is recommended not only in SJBHSI and CHS but also in
other schools whose students face similar challenges in solving word problems related to angles of elevation and
depression. This inclusive approach can effectively address common error patterns and contribute to improved
performance in trigonometry.
5. The researcher recommends for the teacher-made material to undergo quality assurance by the Learning Resource
Management Division (LRDMC) before it is utilized. The researcher also recommends the enhancement of the material
for the utilization in Department of Education.
Acknowledgement
The researcher would like to deliver the gratitude to the Department of Science and Technology – Science Education Institute
(DOST-SEI) for granting me a scholarship under the Capacity Building Program for Science and Mathematics Education
(CBPSME) to pursue this master’s degree program. The researcher also would like to thank the support of the teachers and
students from the SJBHSI and CHS for the opportunity to do research.
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www.ijltemas.in Page 229
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APPENDIX A
Summative Test in Trigonometry
Name:
Score:
Grade and Section:
Date:
Direction: Choose the correct answer from the given options and then write the letter of your answer on the line provided before
the number. You may use your digital scientific calculator to answer the items that requires solving. Good Luck!
______1. With respect to the given angle, what is the ratio of the opposite side to the hypotenuse?
a. sine b. cosine c. secant d. tangent
______2. Which of the following is true about six trigonometric ratios?
a. secant is the reciprocal of sine c. secant is the reciprocal of cosecant
b. cotangent is the reciprocal of cosine d. cosecant is the reciprocal of sine
______3. Which of the following is a secondary trigonometric ratio?
a. tangent b. cosine c. secant d. sine
______4. What is the abbreviation for cosecant?
a. cos b. csc c. sin d. cot
______5. Determine the correct formula for the secant ratio of M.
a.
c.
b.
d.
For items 6-8, refer to the figure at the right
______6. What particular trigonometric ratios can be used to solve for ?
a. sine b. cosine c. tangent d. secant
______7. Which of the following statements is correct?
a.
c.
b.
d.
______8. What is the length of in ?
a. 4.61 units b. 5.61 c. 6.61 d. 7.61
______9. Given where is a right angle, which side is opposite to ?
a.
b.
c.
d. none of the above
______10. Which of the following statements is true about a 45°-45°-90° special triangle?
a. The length of the shorter leg is √3 times the length of the longer leg.
b. The length of the hypotenuse is twice the length of the shorter leg.
c. Both legs of the triangle are equal in length.
d. All of the above.
______11. What is the exact value of ?
a. 1 b. 2 c. 3 d. 4
______12. The value of is _____________ the value of .
a. less than b. greater than c. equal to d. not equal to
______13. Which among the following is the value of ?
A
B
C
60°
8
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a.
b.
c.
d.
______14. Evaluate the trigonometric expression
.
a.
b.
c. 2 d. 1
______15. Determine the value of
.
a. 2 b.
c.
d.
For items 16-18, refer to the figure at the right.
______16. Identify the angle illustrated in the figure.
a. angle of elevation b. angle of depression
c. line of sight d. none of the above
______17. What is the name of the figure formed in the figure.
a. b. c. d.
______18. What is the line of sight in the figure?
a.
b.
c.
d.
______19. This is the angle between the horizontal line of sight and the line of sight up to an object.
a. angle of elevation b. angle of depression
c. line of sight d. none of the above
______20. How can you find the cosecant on a digital scientific calculator?
a. Press the shift button then the sine button.
b. Press the 1, then division operation button and then sine button.
c. Press the inverse button for cosecant.
d. It is impossible to determine the cosecant on a digital scientific calculator.
For items 21-22, refer to the figure at the right.
______21. Which of the following trigonometric ratios
can be used to solve for x?
a.
c.
b.
d.
_______22. What is the measurement of ?
a. c.
b. d.
For items 23-24, refer to the figure at the right.
_______23. What trigonometric ratio can be used to solve for x?
a. cosine c. cosecant
b. sine d. tangent
_______24. What is the length of ?
a. 23. 44 units c. 75.76 units
b. 48.01 units d. 83.99 units
______25. What do you call the angle formed by the horizontal line of sight and the line of sight when the object observed is
below the eye?
a. Horizontal line of sight c. Angle of elevation
30
19
44°
77
For items 16-18, refer to the figure at the right.
______16. Identify what angle is illustrated in the figure.
a. angle of elevation b. angle of depression
c. line of sight d. none of the above
______17. What is the name of the figure formed in the figure.
a. b. c. d.
______18. What is the line of sight in the figure?
a.
b.
c.
d. none of the above
K
L
M
N
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b. Angle of depression d. All of the above
______26. What do you call the angle formed by the horizontal line of sight and the line of sight when the object observed is
above the eye?
a. Horizontal line of sight c. Angle of elevation
b. Angle of depression d. All of the above
______27. Nick is watching on the ground looking up at a concert. What angle can you measure?
a. Angle of elevation c. Horizontal line of sight
b. Angle of depression d. Line of sight
______28. What is NOT true about the illustration at the right?
a. The angle of elevation from point to the plane is 23°.
b. The angle of depression from the plane to point is 23°.
c. The altitude of the plane is 1300 meters.
d. The distance from point to the plane is 3071 meters.
______29. Given the figure below, which statements shows the correct relation of the angle of depression and the angle of
elevation?
a. angle of depression is greater than angle of elevation
b. angle of elevation is greater than angle of depression
c. angle of elevation is not equal to angle of depression
d. angle of elevation is equal to angle of depression
For items 30-32, refer to the figure at the right.
______30. The unknown is an example of _______________.
a. Horizontal line of sight c. Angle of elevation
b. Angle of depression d. All of the above
______31. Which of the following is the most appropriate
trigonometric ratio that relates to the given and unknown?
a.
c.
b.
d. Insufficient data
______32. What is the measurement of ?
a. c. 57.77
b. 56.77 d. 58.77
For questions 33 – 35, use the figure at the right.
______33. What can you infer from the illustration?
a. The angle of depression from the boat to the diver is 16°.
b. The angle of elevation from the diver to the boat is 54°.
c. The distance between the boat and the diver is 54 units.
d. The distance between the diver and the surface of water is the unknown.
______34. What trigonometric ratio can be used to solve for the unknown?
a. sine b. cosine c. tangent d. secant
______35. What is the measurement of the unknown?
8
15
54
16
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a. b. c. d.
For items 36-39, refer to the given problem below.
The angle of elevation of the top of the flagpole is 42° from a point 18 ft away from base.
______36. What trigonometric ratio will be used to find the height of flagpole?
a. sine b. cosine c. tangent d. cosecant
______37. What is the height of flagpole?
a. 16.21 ft b. 3.06 ft c. 8.37 ft d. 14.2 ft
______38. What trigonometric ratio will be used to find the distance of the top of the flagpole from the point?
a. sine b. cosine c. tangent d. cosecant
______39. Calculate the distance of the top of the flagpole from the point?
a. 4.6 b. 15.79 c. 9.98 d. 24.22
______40. Which of the following best describes oblique triangle?
a. All angles are acute c. One angle is right
b. One angle is obtuse d. It has no right angle
______41. What case can be solved by the law of cosines?
a. Two sides and an angle opposite one of them are known
b. Two angles and non-included side are known
c. Three angles are known
d. Three side are known
______42. Which of the following cases can be solved by the law of sines?
a. ASA b. SSS c. SAS d. All of the above
______43. Which of the following is an equation of law of cosine.
a.
c.
b.
d.
______44. Referring to the figure at the right, which angle can you can you say for certain is the largest even before solving the
triangle?
a. A c. C
b. B d. insufficient information
For items 45-50, refer to the given problem below.
Towers A and B are located 250 km apart. A guard in tower A spots a fire at C and sees A to be 50°. Another guard in
tower B finds B to be 42°. How far are the towers from the fire?
______45. Which among the figures illustrates the problem?
a. c.
b. d.
A
B
C
20
22
24
A
B
C
250
50°
42°
A
B
C
250
50°
42°
A
B
C
250
50°
42°
A
B
C
250
50°
42°
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIV, Issue III, March 2025
www.ijltemas.in Page 234
______46. Which of the following cases can best described the figure?
a. ASA b. SAS c. SSS d. AAS
______47. What is the measurement of ?
a. b. c. d.
______48. Which of the following can be used to solve for unknown?
a. Law of Sine b. Law of Cosine c. Law of Tangent d. All of the above
______49. How far is Tower A from the fire?
a. 167.38 km b. 175.25 km c. 191.63 km d. 201.5 km
______50. How far is Tower B from the fire?
a. 167.38 km b. 175.25 km c. 191.63 km d. 201.5 km
APPENDIX B
Worksheet on Angles of Elevation and Depression
Dear Student,
Please consider each word problem carefully. Your answer in this worksheet will be a great help to for identifying the students’
errors when solving word problems involving angles of elevation and depression. This will measure your problem solving skill in
angles of elevation and depreesion. The data to to be gatheredwill be used in my thesis entiled “Students’ Errors in Solving Word
Problems Involving Angles of Elevation and Depression and Performance Level in Grade 9 Trigonometry: Basis for
Development of a Teacher-Made Material”.
Rest assured that your responses in this worksheet will be kept confidential.
Thank you and God Bless!
-RESEARCHER-
Name:
Score:
Grade and Section:
Date:
Directions: Understand the following problems. Draw first the appropriate illustration for each item. Write a trigonometric
equation which can be used to solve the problem. Rewrite the equation until it is calculator-ready, and then solve. Show your
complete solution in the given box below each item. Good Luck! 😊
1. A wire is attached to the top of a 100-meter tower. If the angle of elevation to the top of the tower is , what is the length of
the wire?
2. The angle of depression of a car from the top of a 125-ft tower is . How far is the car from the tower’s base?
3. A 4-meter stick leans against a wall and the base of the stick is 2 meters from the base of the wall. What is the angle of
elevation of the stick.
4. The angle of depression of a boat from the top of a lighthouse is . What is the height of the lighthouse given that the
distance from the boat to the foot of the lighthouse is 255 feet?
5. A diver swims at a depth of 16 feet below sea level. If the direct distance of the diver to a boat is 54 feet, what is the angle of
depression of the boat to the diver?
APPENDIX C
List of Interview Guidelines
Adapted from Rohmah & Sutiarso (2017)
The researcher will use semi-structured interviews as a tool in this study to discover the causes of students' committing such
errors from the perspective of the students themselves. Interview questions consist of structured questions and unstructured
questions (Mahmud & Yunus, 2018). This form of the interview has been classified as an in-depth interview, which is more
adaptable in its application than structured interviews. Newman's Error Hierarchy Model (1977), which sought to identify the
categories of errors made by the students on this subject, was used to conduct the interviews. There will be also follow-up
questions which will be based on the student's responses on the worksheet.
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIV, Issue III, March 2025
www.ijltemas.in Page 235
Problem-Solving
Aspect Of Newman
Interview Questions
Reading Errors
1) Can you read this problem?
2) What information do you get after reading this problem?
3) What are the mathematical symbols contained in this problem?
Comprehension Errors
1) Can you identify the known and the given in this problem?
2) Can you identify where is the error you did?
3) Is there a difficulty in determining what is known and unknown in this problem?
Transformation Errors
1) What mathematical equation did you apply when answering this problem?
2) Can you explain to me how you arrived at this mathematical equation in this
problem?
3) What should be the mathematical equation for this problem?
Process Skills Errors
1) Can you explain how did you solve this problem?
2) Why didn’t you finish solving this problem?
3) Do you have any difficulty in performing the calculation process in the solution
part?
Encoding Errors
1) What conclusion did you draw from this problem?
2) What should be the unit required to this problem?
3) Do you have any difficulty in determining the final answer of this problem?
