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transcript
Running head: ALGEBRA 2 ASSESSMENT
Psychometric Properties of a Computer-Based Algebra 2 Assessment
Michael Mazzarella
George Mason University
ALGEBRA 2 ASSESSMENT
Psychometric Properties of a Computer-Based Algebra 2 Assessment
Research Questions
The purpose of this study is to analyze the psychometric properties, including validity,
Classical Test Theory analysis, and Rasch analysis, of the Mazzarella (2015) Algebra 2
assessment. The following research questions will address that purpose:
1. What are the psychometric properties, particularly validity, of a computer-based Algebra
2 assessment?
2. Are there differences in these properties between the different versions of the assessment
based on item format and question type?
Method
Participants
The participants of this study will be 148 high school students (n = 148) currently
enrolled in an on-level Algebra II course in a large suburban school district during the 2014-2015
school year. These students will be selected based on a sample of convenience. The 148 students
will come from a total of nine on-level high school Algebra II classes, four of which are taught
by the researcher and five of which are taught by three other teachers in the same school. The
three teachers assisting the researcher with this study are on-level Algebra II teachers at the same
school. These teachers have two, seven, and twenty years of experience teaching within the same
school, while the researcher has four years of teaching experience within the same school. The
present study has been fully explained to the other three teachers, and they have agreed to help in
the data collection process and allow their students to participate in the study.
The sample will consist of 61 females (n = 61) and 87 males (n = 87). The ages of the
students at the time of the study will range from fifteen years to nineteen years. The ethnicity of
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the students is expected to be similar to that of the school demographic: About 33% of students
will be Hispanic, 25% of students will be Asian, 24% of students will be white (not of Hispanic
origin), 18% of students will be African-American (not of Hispanic origin), and 2% of students
will be listed as “other.” Approximately twenty students will be categorized as limited English
proficiency. About 55% of students will receive free or reduced lunch on a daily basis.
Approximately 5% of students will be categorized as special education. The school at which the
data is being collected is a public school within a large, diverse, suburban community. The
school has approximately 2,300 students, and the ethnicity of those students will be
approximately the same as that of the sample (source not cited to ensure confidentiality).
Measure
Students will be given a set of thirty mathematics items on the computer through the
program Horizon, which is a commonly used computer-based assessment program in the county
in which the present study will be being conducted. This program allows teachers to create items
of different formats, and teachers will receive the students’ results for each question when the
assessment is complete. The items for this measure can be found in Appendix A.
The curriculum used to create these mathematics items aligns with the standards set forth
by the county in which this study will be conducted. These standards also match the standards
used to create the end-of-year state assessment that students in Algebra II are required to take.
Some examples of standards included in the mathematics measure are solving radical, absolute
value, and rational equations, finding the domain and range of various functions, simplifying
rational and radical expressions, identifying properties of a normal distribution, and recognizing
and solving permutations and combinations. According to the state in which the study will be
taking place, there are four strands (i.e. standards) that categorize Algebra II test questions:
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Expressions and Operations, Equations and Inequalities, Functions, and Statistics (Virginia
Department of Education, 2012). It is also worthwhile to note that Sstudents in the county are
required to complete and pass Algebra II in order to graduate. Thus,This is important to keep in
mind because it implies that students with a variety of skill levels will be measured in the present
study, rather than only students who choose to enroll in an Algebra II course without it being
required.
Procedure and Research Design
In order to obtain assent from students and consent from the students’ parents or
guardians, the researcher will speak with all classes about the purpose of this study, the measure
being used, the students’ optional participation in this study, and the confidential data being
collected. Students will then be given the consent and assent forms, and they will be instructed to
read and sign the assent forms and have their parents read and sign the consent forms. Once the
forms have been read and signed, the students will be instructed to return both forms to their
teacher. The signed forms will be kept in a secure file cabinet in the researcher’s room until the
study has been completed.
After the teacher has received the signed consent and assent forms, students will then
take a measure of mathematics achievement. Students will have one full class period (90
minutes) to complete this test. Once students complete the test, the responses and scores will be
linked to a student ID number that matches that written on the survey. A teacher other than the
researcher will match the responses of the survey and the math measure, enter the data into a
Microsoft Excel file, replace the student ID numbers with a different unidentifiable ID number,
and give the data to the researcher. As an incentive, all students participating in the study will be
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entered into a raffle to win one of four $25 Target gift cards. Each of the three teachers assisting
with the study will also receive a $25 Target gift card.
Classes will be randomly assigned to take one of four versions of the math measure: Test
A, Test B, Test C, or Test D. Test A and Test B have the same prompts in the same order. On
Test A, the odd-numbered questions will be multiple choice questions, and the even-numbered
questions will be technology-enhanced questions (either fill-in-the-blank or selecting multiple
correct answers). On Test B, the even numbered questions will be multiple choice questions, and
the odd numbered questions will be technology-enhanced questions. Test questions are staggered
this way so that data will be collected for every prompt in both a multiple choice and technology-
enhanced format. On both Test A and Test B, there will be twenty straightforward mathematics
problems (ten multiple choice and ten technology-enhanced) and ten real-world application
problems (five multiple choice and five technology-enhanced). Creating the assessments in this
way yields four categories: straightforward multiple choice, straightforward technology-
enhanced, word problem multiple choice, and word problem technology-enhanced. Both tests
have the same number of questions in each category. These tests were created in this way
because the data collected will provide a comparison of the same prompts with different item
formats, as well as a comparison between straightforward questions and word problems. The
items in Test A and Test B can be found at the end of this document in Appendix B.
Test C will have the same exact questions and formats as Test A, but the questions will
be in reverse order. Test D will have the same exact questions and formats as Test B, but the
questions will be in reverse order. Test C and Test D were created to account for test fatigue.
Tyrrell and colleagues (1995) support that visual and mental fatigue can occur as students take
assessments, especially when the assessments are taken on the computer. As students work
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through the thirty-question math measure, some may become less motivated or energized toward
the end of the test. Thus, it is important to test whether or not this occurs in the measure before
drawing conclusions about specific test questions or overall achievement.
As mentioned previously, the results of the mathematics measure will be available online
to the student’s teacher once the test is complete. Results will include the number of correct
answers and the responses to each multiple choice and technology-enhanced question.
Data Analysis
All data analysis will be conducted using the computer programs Microsoft Excel, SPSS,
and jMetrik. In this study, the data analysis will attempt to simultaneously answer both research
questions. The first research question asks: What are the psychometric properties, particularly
validity, of a computer-based Algebra 2 assessment? The second research question asks: Are
there differences in these properties between the different versions of the assessment based on
item format and question type? In order to answer these questions, several methods of analysis
will be used. First, two independent samples t-tests for item order will be used to determine if
there are significant differences in the results of Test A and Test C, or the results of Test B and
Test D. As mentioned before, Test A and Test C have the same exact prompts and item formats,
but are in reverse order, and Test B and Test D follow the same format. By comparing the results
of the two pairs of versions, the researcher will determine if there is an ordering effect for the
assessment.
Second, Classical Test Theory (CTT) analysis will be run. Several descriptors of each
item will be looked at. These descriptors include reliability, which is defined as the ratio of true
score variance to observed score variance; item difficulty, which is defined as the proportion of
examinees who answered the item correctly; and item discrimination, which is defined as the
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difference between the proportion of examinees in the upper group who answered the item
correctly and the proportion of examinees in the lower group who answered the item correctly
(Osterlind, pp. 122, 277-288). These CTT analyses will be applied to both student responses to
the individual items, as well as the results of the overall assessment.
Third, Rasch analysis will be applied to the data. This analysis will be used to indicate
several more characteristics about the items and the assessment. One such characteristic is fit of
items. The program jMetrik will identify misfit items, which could be an indication that the item
does not discriminate well between high and low achieving students, the item is general not a
good fit in the assessment as compared to other items, or an anomaly in the examinee’s
responses. Rasch analysis using jMetrik will also indicate the person separation index and item
separation index. The person separation index will indicate whether or not the assessment
adequately distinguishes between high and low achieving individuals, while the item separation
index will indicate whether or not the assessment adequately distinguishes between high and low
difficulty items (Dimitrov, 2014). Rasch analysis will also give three-parameter logistic (3PL)
models for each multiple choice item. The 3PL models are visual representations of student
responses based on the item discrimination, the item difficulty, the chance of randomly guessing
correctly, and the student ability level, which is based on the number of standard deviations the
students’ achievement is from the mean of a normal curve (Birnbaum, 1968; Harris, 1989).
When the 3PL model is created for multiple choice items, the chance of randomly guessing the
item correctly is .25. The researcher will be looking for curves that do not follow the normal
shape of a 3PL model or curves that are drastically different than the other curves, which may
indicate misfit items.
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Fourth, a Wright map will be constructed. A Wright map, also known as an item-person
map, is used to show the distribution of examinees based on ability level, alongside the
distribution of items based on difficulty. The purpose of the Wright map is to visualize these
distributions and compare them to one another. Ideally, the two distributions will be similar in
nature with the peak of each around the same part of the axis. A Wright map could also be used
as a means of reducing the number of items in an assessment, but it will not be used in that way
for the present study because each item measures a different part of the Algebra 2 curriculum
(Neumann et al., 2010; Kuo et al., 2015). The researcher will construct a Wright map for each
test version.
Finally, based on the previous analysis, the researcher will go in depth on each item that
appears to be a misfit, determine some initial effects of item format and question type, and draw
some overall comparisons and observations about each test version. By doing so, the researcher
will attempt to validate that this Algebra 2 assessment is valid and an acceptable measure for
future studies involving the same mathematical content.
Results
At the end of the study, 41 students took Test A, 41 students took Test B, 45 students
took Test C, and 21 students took Test D. Two independent samples t-tests were run to determine
if there was an ordering effect for Tests A and C, and for Test B and D. The t-test for Tests A
and C, which did not have equal variances according to Levene’s Test, showed that t(76.416) = -
3.75, p < .001. The t-test for Tests B and D, which had equal variances according to Levene’s
Test, showed that t(60) = -2.07, p < .05. Thus, both tests showed an ordering effect for the
separate versions. However, upon further examination, it was determined that one teacher’s
classes significantly skewed the data; this teacher’s students achieved significantly lower than
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the students of the other two teachers. When this teacher’s students were removed from the tests,
the t-tests for Tests A and C and Tests B and D showed that t(58,746) = -0.48, p > .05, and t(43)
= -0.53, p > .05, respectively. Thus, it is recommended by the researcher that the data for Tests A
and C and for Tests B and D can be combined, resulting in two versions: Test A and Test B.
When combined, a total of 86 students (n = 86) took Test A, while 62 students (n = 62) took Test
B.
Classical Test Theory analysis was run on the data. As mentioned before, half of the
items on each version were multiple choice items and half of the items were technology
enhanced (either multiple select or fill-in-the-blank). Table 1 shows results of CTT analysis. On
Test A, the mean score was 17.0233 (out of 30 questions). The skewedness and kurtosis values
were -0.3956 and -0.6510, respectively, indicating that the sample is normally distributed. The
overall reliability of Test A, according to the Cronbach’s alpha value, was α = .8815. The
standard error of measurement (SEM) was 2.2417. The jMetrik program also indicates the
change in reliability if any of the items were removed from the data. The Cronbach’s alpha value
would be no larger than α = .8839, and therefore it is not recommended that any of the items be
removed from this version. On Test B, the mean score was 18.3871. The skewness and kurtosis
values were -0.1075 and -0.6814, respectively, indicating that the sample is normally distributed.
The Cronbach’s alpha value of Test B was α = .7761. It is also not recommended that any items
be removed from Test B because the Cronbach’s alpha value would be no larger than α = .7903.
The difficulty of the items on Test A ranged from .2093 (Question #17) to .8837 (Question #7),
while the difficulty of the items on Test B ranged from .1774 (Question #3) to .9516 (Question
#5 and Question #10). The discrimination of the items on Test A ranged from D = .1123
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(Question #17) to D = .6226 (Question #25), while the discrimination of the items on Test B
ranged from D = -.1115 (Question #2) to D = .5279 (Question #25).
Test A Test B
n 86 62
Mean 17.0233 18.3871
Median 19 19
Standard Deviation 6.5129 4.8704
Skewness -0.3956 -0.1075
Kurtosis -0.6510 -0.6814
Cronbach’s Alpha 0.8815 0.7761
Table 1. CTT analysis of both test versions
Rasch analysis was then run on jMetrik. The standard weighted mean squares and
standard unweighted mean squares values will indicate the any misfit items in the tests.
According to this output, Test A had three potential misfit items (Question #17, Question #19,
and Question #29), and Test B also had three potential misfit items (Question #2, Question #4,
and Question #28). The item separation index of Test A is 4.1273, while the item separation
index of Test B is 3.4176. This indicates that both versions of the test can moderately but
acceptably distinguish between item difficulties. The person separation index of Test A is
2.5484, while the person separation index of Test B is 1.8852. The 3PL Rasch analysis models
for each item were run, and the most notable outcomes are displayed in Figure 1. Finally, the two
Wright maps were run for Test A and Test B, and can be found in Figure 2 and Figure 3,
respectively.
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INSERT FIGURE 1 HERE
Discussion
The purpose of this study was to analyze the items and results of the Algebra 2
assessment given to students in a large, diverse, suburban school district. The results of the
analysis show insight into the nature of the items based on item format or question type, or
differences in the two test versions. Both test versions were found to be fairly reliable (α = .8839
and α = .7761). It is worthy to note that the reliability of Test B is likely lower than that of Test
A due to the smaller number of students who took Test B. Nevertheless, both versions have an
acceptable level of reliability. Both versions had two questions with a difficulty less than .25. On
Test A, one of these questions was a straightforward multiple choice item (Question #17) and the
other was a multiple select word problem item (Question #24). On Test B, one question was a
straightforward multiple select item (Question #3) and the other was a multiple select word
problem item (Question #29). Further research is necessary to determine if there are significant
differences between achievement on different item formats and question types.
It is also worthwhile to discuss specific items with notable discrimination values. The
lowest discrimination of either version is Question #2 on Test B. This item is a straightforward
multiple choice question asking students to find a specific term in a recursive sequence. The item
has a negative discrimination, which is problematic because it indicates that students in the
lowest ability group answered this question more accurately than students in the highest ability
group. However, it should be noted that the content in this item was one of the last topics to be
taught in the Algebra 2 curriculum, so it is possible that one or more teachers did not fully teach
topic at the time of the study. The lowest discrimination item on Test A was Question #17. This
item, which was a straightforward multiple choice item, asks students to select the correct end
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behavior of a rational function. Based on the data, there seems to be an observable relationship
between discrimination and item format; that is, low discrimination items seem to be more often
than not multiple choice items. This additional observation supports the need for further research
regarding item format.
INSERT FIGURE 2 HERE
As mentioned previously, each version had three potential misfit items, but none of the
prompts are misfits on both versions. This suggests that none of the prompts themselves are the
cause of the items being misfits. The strongest misfit on either version is Question #2 on Test B.
The cause of this item is likely the negative discrimination, which was discussed in the previous
paragraph. Due to the large misfit value, it is recommended that this item be removed from the
analysis.
DISCUSSION OF 3PL MODELS
INSERT FIGURE 3 HERE
DISCUSSION OF WRIGHT MAP
Limitations
There are several limitations of this analysis that must be considered. First, the results of
the study cannot be generalized in several ways. Socioeconomically, the school was in a large
suburban school district with a number of resources to provide students. The results of this study
may be different if it were replicated in an urban or rural school setting. Additionally, the
students who participated in this study were in an on-level algebra class with a specific
curriculum and pacing guide. This prevents the researcher from stating that these results would
be similar in an honors or below-level setting. Finally, since the material in this assessment was
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from an Algebra 2 curriculum, one cannot assume that similar psychometrics would occur for a
geometry test, for example.
Another limitation of this study is the imbalance of participants who took each type of
test. Based on the procedures of the study and the abilities of the software on which the test was
administered, each class had to take the same test version. Since there were an odd number of
classes who participated in the study, there were more students that took Test A than those who
took Test B. This provided a limitation because some statistical results may have been altered.
For instance, Test A had a higher reliability than Test B, which may have been caused by the
lower number of students who took Test B. Also, the separation indices for Test B were
unfavorably low, most likely due to a low number of participants. The researcher suggests
replicating this study with a larger number of students to determine if these statistics are a result
of sample size.
Next, the fact that students came from three different teachers also causes a limitation. In
particular, one teacher’s students scored significantly lower than the students of the other two
teachers. This was problematic in that these students’ scores were inconsistent with the scores of
other students. Having students from different teachers participate can also generally cause some
noticeable differences. For example, some teachers may be more effective at teaching some
material than others, but ineffective at teaching other content. Additionally, the pacing of some
teachers may be different based on their teaching styles or how they feel their students are
performing. It is suggested that additional studies be conducted using students from only one
teacher.
Finally, further research is needed to determine if there are significant differences in
items based on item format or question type. The present study made some general observations
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regarding these variables, but more analysis is needed to study to what extent they impact student
achievement. In addition to simply analyzing the results of the different item format and question
types, it is also important to study what motivational constructs impact achievement on these
items. Characteristics such as motivation, task value, and interest all have been shown to increase
math achievement (Wigfield, Tonks, & Eccles, 2004); however, do these constructs predict
achievement on each item type differently? Further research is needed to answer this question.
Finally, gender differences were not explored in the present study. Additional research should
compare the assessment results based on gender, both at an overall level and looking only at
specific types of prompts and entry methods.
Educational Implications
Given the current emphasis on standardized testing, theThe results of this study carry
strong educational implications, especially in an era in which there is so much emphasis on
standardized testing. TheAs mentioned previously, the questions on the Mazzarella (2015)
assessment used in this study were modeled of the Virginia Algebra 2 SOL exam. Therefore,
these results can be used to assessdetermine whether or notthe reliability and validity of this type
of exam and identify potential issues. is valid and fair. This is immensely important because of
the Vvast number of students who take this difficult standardized test every year. For teachers,
these results can be used to help understand what students struggle with most, whether it is a a
substantive factor such as a particular mathematical skill or concept, the way a question was
asked, or the way that students enter their answer. For test makers, these results can help shape
the way that statewide assessments are created. If items are found to be legitimate misfits, it is
important for test makers to determine why they are misfits and how they can be improved.
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Psychometrics studies like the present study should constantly consisently be conducted so that
tests are always being scrutinized and improved.
Overall, the present study sheds light on the characteristics of a specific computer-based
mathematics assessment. The in-depth features of the test and its items provide valuable
information to teachers, students, test makers, and researchers. It is the hope of the researcher
that this study will be used to create better tests, lead to more research on computer-based math
assessments, and ultimately lead to fair tests that accurately represent students’ understanding
and can help them meet their full potential.rultimately help students succeed to their full
potential.
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References
Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee’s ability. In
F.M. Lord and M.R. Novick (Eds.), Statistical theories of mental test scores. Reading,
MA: Addison-Wesley.
Dimitrov, D. (2014). Topic 4 – Classical Test Theory [PowerPoint Slides]. Retrieved from
https://mymasonportal.gmu.edu.
Dimitrov, D. (2014). Topic 8 – Item Response Theory Part 4: Rasch model- Data fit, item/test
information, and jMetrik Rasch Analysis [PowerPoint Slides]. Retrieved from
https://mymasonportal.gmu.edu.
Harris, D. (1989). Comparison of 1-, 2-, and 3-parameter IRT models. Educational
Measurement: Issues and Practice, 8(1), 35-41. doi:10.1111/j.1745-3992.1989.tb00313.x
Kuo, C., Wu. H., Jen, T., & Hsu, Y. (2015). Development and validation of a multimedia-based
assessment of scientific inquiry. International Journal of Science Education, 37(14),
2326-2357. doi:10.1080/09500693.2015.1078521
Mazzarella, M. (2015). Algebra II Review. Unpublished mathematics assessment, George Mason
University, Fairfax, VA.
Neumann, I., Neumann, K., & Nehm, R. (2010). Evaluating instrument quality in science
education: Rasch-based analysis of a nature of science test. International Journal of
Science Education, 33(10), 1373-1405. doi:10.1080/09500693.2010.511297
Osterlind, S. J. (2010). Modern measurement: Theory, principles, and applications of mental
appraisal. Boston, MA: Allyn & Bacon.
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Tyrrell, R., Holland, K., Dennis, D., & Wilkins, A. J. (1995). Coloured overlays, visual
discomfort, visual search and classroom reading. Journal of Research in Reading, 181,
10-23.
Virginia Department of Education (2015). Standard of learning (SOL) and testing. Retrieved
from Virginia Department of Education website: http://www.doe.virginia.gov.
Wigfield, A., Tonks, S., & Eccles, J. S. (2004). Expectancy value theory in cross-cultural
perspective. In D. M. McInerney & S. Van Etten (Eds.), Big theories revisited (pp. 165-
198). Greenwich, CT: Information Age.
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Appendix A: Mathematics Measures
Test A
Question 1 :What is the solution set for this equation?
A:
B:
C:
D:
Question 2 :Type your answer into the box. You must give your answer in integer form.
The following sequence is given in recursive form.
What is the value of the fourth term of this sequence?
Question 3 :Which of the following situations involves a permutation?
A: Determining how many different groups of 3 employees can be chosen from 9 employees.
B: Determining how many different ways 7 runners can be assigned lanes on a track for a race.
C: Determining how many different ways to choose 10 students to attend a field trip from a group of 25 students.
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D: Determining how many different ways 4 cashiers can be chosen to work from a group of 7 cashiers.
Question 4 :Click on a box to choose each y-coordinate you want to select. You must select all correct answers.
What are the y-coordinates for the solution to this system of equations?
A: y = -9
B: y = -3
C: y = -2
D: y = 1
E: y = 2
F: y = 6
G: y = 8
H: y = 9
Question 5 :The number of combinations of 7 objects taken 2 at a time is
A: 3B: 7
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C: 21D: 42
Question 6 :Type your answer into the box. You must enter your answer in integer form.
The shoe sizes of a large population are normally distributed with a mean of 8.9 inches and a standard deviation of 0.705 inches. What percentage of the population has a shoe size greater than 9.8 inches? ROUND TO THE NEAREST INTEGER.
Question 7 :
Factor:
A: (2x+3) (3x-7)
B: (2x-3) (3x+7)
C: (3x+2) (2x-7)
D: (3x-2) (2x+7)
Question 8 :Select a box for each correct part of the expression. You must select each correct expression.
Select each part of the simplified expression .
A:
B:
C:
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D:
E:
F:
Question 9 :The area of a triangle varies jointly with the product of the base and the height. A triangle has a base of 12 feet, a height of 3 feet, and an area of 18 square feet. What is the base of a triangle with a height of 4 feet and an area of 36 square feet?
A: 0.5 feet
B: 9 feet
C: 12 feet
D: 18 feet
Question 10 :Type the answer into the box.The number of permutations of 9 objects taken 3 times is
Question 11 :
Which is a solution of ?
A: x = -5
B:x
= C: x = -
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1
D: x = 1
Question 12 :Click on the box to select the value. You must select each correct value.
Two baseballs were thrown on a field as the same time. One ball follows the
path of the function , and the other ball follows the path of
the function , where x is the time in seconds, and f(x) and g(x) are the heights in feet. At what two times, in seconds, are the two balls the same height?
A: 0.7 seconds
B: 1.0 seconds
C: 4.0 seconds
D: 5.14 seconds
E: 5.45 seconds
F: 7.13 seconds
Question 13 :What is the sum of this infinite series?
72 - 36 + 18 - 9 + ...
A: -144
B: -48C: 48
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D: 144
Question 14 :Type your answer into the box. You must enter your answer in integer form.
Let and , what is ?
Question 15 :A new rollercoaster at an amusement park follows the path of the
function , where x is the time, in seconds, after the rollercoaster begins, and f(x) is the height of the rollercoaster, in yards. Between which two times, in seconds, is the rollercoaster increasing in height?
A:
B:
C:
D:
Question 16 :Select each expression that is equivalent. You must select all correct expressions.
Identify each expression that is equivalent to 1.
A:
B:
C:
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D:
E:
F:
G:
H:
Question 17 :
Which of the following describes the end behavior of as x approaches infinity?
A: y approaches negative infinity
B: y approaches -2
C: y approaches 3
D: y approaches infinity
Question 18 :Type your answer into the box. You must enter your answer in integer form.
The heights of Galapagos penguins are normally distributed with a mean of 49 cm and a standard deviation of 1.82 cm. If a scientist measures the heights of 300 penguins, how many penguins are expected to be between 48.4 cm and 50.1 cm tall? ROUND YOUR ANSWER TO THE NEAREST INTEGER.
Question 19 :Which is the equation of an asymptote of the graph of the following equation?
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A: x = -3
B: y = 3
C: x = 6
D: y = 6
Question 20 :Click on the box to select an interval. You must select each correct interval.
Indicate each intervals where the graph is only increasing.
A:
B:
C:
D:
E:
F:
G:
H:
Question 21 :A mathematics class consists of 10 girls and 8 boys. The teacher
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wants to choose 2 girls and 2 boys to go on a trip. How many different groups could the teacher choose?
A: 73
B: 146
C: 1260
D: 5040
Question 22 :Type your answer into the box. You must enter your answer in integer form.
If y varies directly with the square root of x, what is the constant of variation if y = 36 when x = 9?
Question 23 :
What are the zeros of the function ?
A: x = -16 and x = 0
B: x = -16, x = -8, and x = 2
C: x = -8 and x = 2
D: x = -2 and x = 8
Question 24 :Click on the box to select the correct equation. You must select each correct equation.
A baseball was thrown by a player, and hit the ground after exactly 5 seconds. If x represents the time in seconds and y represents the height of the ball, which of the following functions could represent the path of the
ALGEBRA 2 ASSESSMENT
ball?
A:
B:
C:
D:
E:
F:
Question 25 :A normally distributed set of numbers has a mean of 75 and a standard deviation of 7.97. What percentage of values lies between 70 and 85?
A: 11%B: 37%C: 63%D: 89%
Question 26 :Click on the correct box to select each value. You must select each correct value.
The domain of the function is all real numbers except -
A: -7B: -4C: -3D: 0E: 3
ALGEBRA 2 ASSESSMENT
F: 4G: 7
Question 27 :In a school, 12 students are running for 4 class officers- a president, a vice president, a secretary, and a treasurer. If each position is to be held by one person and no person can hold more than one position, in how many ways can those positions be filled?
A: 48
B: 495
C: 11880
D: 20736
Question 28 :Click on each box to select each function. You must select each correct function.
Which of the following functions are in the same family as the
function ?
A:
B:
C:
D:
E:
F:
ALGEBRA 2 ASSESSMENT
Question 29 :The steps to simplify an expression are shown below:
Step 1: 4(x+3) - 3x + 1Step 2: 4x + 12 - 3x + 1Step 3: 4x - 3x + 12 + 1Step 4: x + 13
Which of the following properties justifies getting from Step 2 to Step 3?
A: Associative Property
B: Commutative Property
C: Distributive Property
D: Transitive Property
Question 30 :Type your answer into the box. ROUND YOUR ANSWER TO THE NEAREST TENTH.
A baseball player throws a ball from one end of the field to the other. A fan measures the path of the ball and determines that it follows the
function , where x is the time in seconds and f(x) is the height in feet. What is the maximum height of the ball, in feet? ROUND YOUR ANSWER TO THE NEAREST TENTH.
Test B
ALGEBRA 2 ASSESSMENT
Question 1 :Type your answer into the box. Enter your answer as a whole number.
What is the solution set for this equation?
Question 2 :The following sequence is given in recursive form.
What is the value of the fourth term of this sequence?
A: 29B: 33C: 61
D: 125
Question 3 :Click on the box to choose the situation. You must select all correct situations.
Which of the following situations involve a permutation?
A: Determining how many different ways to choose 3 employees from a group of 9 employees.
B: Determining how many different ways 7 runners can be assigned lanes on a track for a race.
C: Determining how many different seating charts can be made placing 6 people around a table.
D: Determining how many 5-letter passwords can be made using the
ALGEBRA 2 ASSESSMENT
word "graph."
E: Determining how many different groups of 10 students can be chosen to go on a field trip from a group of 25 students.
F: Determining how many different ways 4 cashiers can be chosen to work from a group of 6 cashiers.
Question 4 :What are the y-coordinates for the solutions to this system of equations?
A: y = 1 and y = 9
B: y = -3 and y = -9
C: y = -2 and y = 6
D: y = 2 and y = 8
Question 5 :Type the answer into the box.
The number of combinations of 7 objects taken 2 at a time is
Question 6 :The shoe sizes of a large population are normally distributed with a mean of 8.9 inches and a standard deviation of 0.705 inches. What percentage of the population has a shoe size greater than 9.8 inches? Round to the nearest integer.
A: 5%B: 10%C: 20%
ALGEBRA 2 ASSESSMENT
D: 34%
Question 7 :Click on a box for each factor you want to select. You must select all correct factors.
Select all of the factors of:
A: (2x+3)
B: (2x-3)
C: (3x+2)
D: (3x-2)
E: (2x+7)
F: (2x-7)
G: (3x+7)
H: (3x-7)
Question 8 :
Which of the following expressions is equivalent to ?
A:
B:
C:
ALGEBRA 2 ASSESSMENT
D:
Question 9 :Type your answer into the box. You must enter your answer in integer form.
The area of a triangle varies jointly with the product of the base and the height. A triangle has a base of 12 feet, a height of 3 feet, and an area of 18 square feet. What is the base of a triangle with a height, in feet, of 4 feet and an area of 36 square feet?
Question 10 :The number of permutations of 9 objects taken 3 times is
A: 27B: 84
C: 504
D: 729
Question 11 :Click on each solution to the equation. You must select each correct solution.
Select all the solutions of .
A: x = -5
B:x
=
C: x = -1
D: x = 1
ALGEBRA 2 ASSESSMENT
E:x
=
F: x = 5
Question 12 :Two baseballs were thrown on a field as the same time. One ball
follows the path of the function , and the other ball
follows the path of the function , where x is the time in seconds, and f(x) and g(x) are the heights in feet. At what two times, in seconds, are the two balls the same height?
A: 0.70 seconds and 5.45 seconds
B: 7.13 seconds and 5.14 seconds
C: 0.70 seconds and 7.13 seconds
D: 5.14 seconds and 5.45 seconds
Question 13 :Type your answer into the box.
What is the sum of this infinite series?
72 - 36 + 18 - 9 + ...
Question 14 :
Let and . What is ?
A: -29B: 35C: 75
ALGEBRA 2 ASSESSMENT
D: 152
Question 15 :Click on the box to select the interval. You must select each correct interval.
A new rollercoaster at an amusement park follows the path of the
function , where x is the time, in seconds, after the rollercoaster begins, and f(x) is the height of the rollercoaster, in yards. Between which two times, in seconds, is the rollercoaster increasing in height?
A:
B:
C:
D:
E:
F:
Question 16 :Which expression is equivalent to 1?
A:
B:
C:
D:
ALGEBRA 2 ASSESSMENT
Question 17 :Type your answer into the box. You must give your answer in integer form.
What value does y approach in the function as x approaches infinity?
Question 18 :The heights of Galapagos penguins are normally distributed with a mean of 49 cm and a standard deviation of 1.82 cm. If a scientist measures the heights of 300 penguins, how many penguins are expected to be between 48.4 cm and 50.1 cm tall? Round your answer to the nearest integer.
A: 84
B: 107
C: 168
D: 204
Question 19 :Click on each box to choose each asymptote. You must select all correct asymptotes.
Which are the equations of the asymptotes of the graph of the following function?
A: x = -3
B: y = -3
C: x = 3
ALGEBRA 2 ASSESSMENT
D: y = 3
E: x = 6
F: y = 6
Question 20 :
Throughout which of the following intervals is only increasing?
A:
B:
C:
D:
Question 21 :Type your answer into the box. You must enter your answer in integer form.
A mathematics class consists of 10 girls and 8 boys. The teacher wants to choose 2 girls and 2 boys to go on a trip. How many different groups could the teacher choose?
Question 22 :If y varies directly as the square root of x, what is the constant of variation if y = 36 and x = 9?
A: 1.5
B: 2C: 4D: 12
ALGEBRA 2 ASSESSMENT
Question 23 :Click on the box to select the zeros. You must select each correct zero.
Which of the following are zeros of the function ?
A: x = -16
B: x = -8
C: x = -2
D: x = 0
E: x = 2
F: x = 8
Question 24 :A baseball was thrown by a player, and hit the ground after exactly 5 seconds. If x represents the time in seconds and y represents the height of the ball, which of the following functions could represent the path of the ball?
A:
B:
C:
D:
Question 25 :Type your answer into the box. You must enter your answer in
ALGEBRA 2 ASSESSMENT
integer form.
A normally distributed set of numbers has a mean of 75 and a standard deviation of 7.97. What percentage of values lies between 70 and 85? ROUND TO THE NEAREST INTEGER.
Question 26 :
The domain of the function is all real numbers except -
A: -7, -4, 4
B: -7, 4
C: -4, 7
D: 4
Question 27 :Type your answer into the box. You must enter your answer in integer form.
In a school, 12 students are running for 4 class officers- a president, a vice president, a secretary, and a treasurer. If each position is to be held by one person and no person can hold more than one position, in how many ways can those positions be filled?
Question 28 :Which of the following is in the same family as the
function ?
A:
B:
C:
ALGEBRA 2 ASSESSMENT
D:
Question 29 :Click on the box to select a property. You must select each correct property.
The steps to simplify an expression are shown below:
Step 1: 4(x+3) - 3x + 1Step 2: 4x + 12 - 3x + 1Step 3: 4x - 3x + 12 + 1Step 4: x + 13
Which of the following properties justify Step 2, Step 3, and Step 4?
A: Associative Property
B: Commutative Property
C: Distributive Property
D: Inverse Property
E: Substitution Property
F: Transitive Property
Question 30 :A baseball player throws a ball from one end of the field to the other. A fan measures the path of the ball and determines that it
follows the function , where x is the time in seconds and f(x) is the height in feet. What is the maximum height of the ball, in feet?
A: 2 feet
ALGEBRA 2 ASSESSMENT
B: 5.4 feet
C: 6 feet
D: 9.2 feet