Maggie Sharp
EPSY 485
Assessment during Instruction Assignment
1. School Context
The school I was placed in this semester had the following demographics: 48.4% White, 33.9%
Black, 8.2% Asian, 6.0% Hispanic, 3.2% Multiracial, and 0.2% American Indian. In the school,
47.4% of the student population qualifies as low income and 10.5% of students have IEPs.
Overall, 51.6% of the students meet or exceed standards on all subjects, which is significantly
lower than the district’s 71.7%. Regardless, neither district is meeting AYP as of 2012.
Demographically the school is very comparable to the district as a whole. I was placed in an
Algebra II classroom with 21 students, of which 7 students have serious attendance problems. On
any given day the class has about 12-14 students present. This class is made up of a majority of
low income students. Two of the students have IEPs, one for a math deficit and the other for
behavior issues. The student with a math deficit leaves about 20 minutes early each day to be
bussed to another school for a vocational class. Another student was on an IEP but recently
decided to discontinue services. The class is made up of students ranging from sophomores to
seniors. All of the seniors in the class were re-taking Algebra II after having failed it the year
before. Nine of the students currently have a D or lower in the class, 8 have a B- or higher, and
the other 4 students have a C. Overall, the class performs below average as compared to the
school population.
2. Unit of Study
The Unit we are working on is from Appendix B of the CPM textbook. It is called Exponential
Functions and Models. The intended grade level is Algebra II, and in this case the class consists
of sophomores through seniors. The enduring questions of the unit are as follows:
What properties are characteristic the graphs of exponential functions? What connections
exist between graphs, tables, and equations of exponential functions? What are important
properties of exponents which help us solve exponential equations?
An important enduring understanding is that students know how to create exponential models
from real-life situations both for exponential growth and decay. Another enduring understanding
is to see how different values put into the equation affect the shape of the exponential curve.
I have chosen the following learning objectives from the mathematical Common Core
standards:
1. Students should be able to graph functions expressed symbolically and show key features
of the graph, by hand in simple cases and using technology for more complicated cases.
a. Graph exponential equations showing end behavior and intercept
(Bloom’s: Remember and Apply)
2. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for
specific values of k (both positive and negative). (Bloom’s: Understand)
3. Students should be able to interpret the parameters in an exponential function in terms of
a context. (Bloom’s: Understand and Create)
Maggie Sharp
EPSY 485
3. Activity (with answer key)
1. Graph the following two exponential functions. Be sure to note key features in words
next to or under the graph you draw.
1
𝑦 = ( )𝑥
𝑦 = 4𝑥
6
y-int: (0,1)
Important
points:
(-1, 1/4), (1, 4)
Asymptote: x=0
y-int: (0,1)
Important
points:
(-1, 6), (1, 1/6)
Asymptote: x=0
2. Below is the graph of 𝑓(𝑥) = 2𝑥 . For each change made to 𝑓(𝑥) Please describe how the
graph will change. (Pictures are also acceptable if they are clear and accurate).
a. 𝑓(𝑥) = 2𝑥 + 4
The graph will be moved up in the y-direction
by 4, causing the y-intercept to change from 1
to 5 and the asymptote will be x=4
b. 𝑓(𝑥) = −(2)𝑥
The graph will be reflected over the x-axis so that
each point (x,y) from the original function becomes
(x,-y) in the new function. The asymptote is still
x=0
c. (𝑥) = (2𝑥−3 )
The graph shifts right by 3 units so each point (x,y) in the original function becomes
(x+3,y) in the new function. The asymptote is still x=0
1 𝑥
d. 𝑓(𝑥) = ( )
2
The graph will be reflected over the y-axis so that each point (x,y) from the original
function becomes (-x,y) in the new function. The asymptote is still x=0.
Maggie Sharp
EPSY 485
3. Tickets for a concert have been in incredibly high demand, and as the date for the concert
draws closer, the prices of tickets increases exponentially. The cost of a pair of concert
tickets was $150 yesterday, and today it is $162. Assume each day’s percent increase
from the day before is the same.
a. What is the daily percent rate of increase? What is the multiplier?
The daily increase is given by: (162-150)/150 which is 0.08. This gives a daily
percent rate of increase of 8%.
The multiplier is 150 assuming we call yesterday the “initial value” or day 0.
The multiplier is 162 assuming we call today the “initial value” or day 0.
b. Write an equation which models this situation.
𝒚 = 𝟏𝟓𝟎(𝟏. 𝟎𝟖)𝒙
*Must take into account the multiplier used by the students in question a and
appropriately adjust
c. What will be the cost of a pair of concert tickets one week from now? (Assume
now is today)
Plug in 8 for x if the multiplier is 150 and plug in 7 for x if the multiplier is
162.
y = 277.6395…
A pair of tickets will be $277.64.
d. What was the cost of a pair of concert tickets two weeks ago?
Plug in -13 for x if the multiplier is 150 and plug in -14 for x if the multiplier
is 162.
y = 55.1546887…
A pair of tickets would have been $55.15.
Item Analysis:
Item 1:
This item is meant to measure the first learning objective because it asks students to graph the
exponential expressions based on their symbolic equations. It also calls for them to recognize
key features because they are asked to record them with their graph. This will allow me to
measure whether or not my students understand the basic shape an exponential graph will take
with a simple base. Since the equations are simple, it is a good way to see what fundamental gaps
of understanding exist for the students without complicating the expression. This item measures
whether the students remember the basic shapes we see in exponential curves as well as which
key features are important to identify. It also measures if the students can apply the knowledge
of similar examples from previous classes to two symbolic expressions they have not specifically
seen before.
Maggie Sharp
EPSY 485
Item 2:
This item is meant to measure the second learning objective because it asks students to state how
different values of k will affect the picture we see when we graph the function. It allows me to
see whether or not the students understand that placing k in different parts of the function affect
the graph differently. Also, part d measures their understanding of the “partner” exponential
graphs which were discussed the previous day. Students should be able to recognize that when
we graph an exponential function, its reciprocal takes an identical shape but is reflected over the
y-axis. This item measures whether the students understand the symbolic manipulation of an
expression to the transformation of the graph because the students are asked to classify and
explain what kind of transformation is occurring.
Item 3:
This item is meant to measure the third learning objective because it asks students to classify a
real-world situation in terms of an exponential expression. This puts exponential functions into a
new context, but one that students would have been familiar with from previous lessons. By
asking them first to write down the multiplier and percent, it gives them a greater opportunity to
connect their prior knowledge and create the correct formula. This also allows me to see if the
students understand the connection between these terms and the actual formula. Furthermore, the
students are given the opportunity to use the formula in the context of the problem, which
provides another opportunity to demonstrate the connection between the situation and the
equation they wrote. This question requires students to understand how to connect a situation to
an equation and vice versa. This question also requires students to create an expression which
represents the situation.
I have made an effort to create items that are fair by using wording that does not present any
biases based on culture or background. Furthermore, I have provided clear directions when there
may be confusion. For example, in Item 1, I want students to actually write out the key feature so
I stated that they must write them “in words.” I have seen many students lose credit for just
pointing them out on the graph, and if I had not specified this would not be a fair point
deduction. I also provided multiple options for an answer in the last 2 items. For item 2, I noted
that students could draw pictures instead of describing. If the picture is accurate, this is an
equivalent answer in terms of understanding, and it may help students with limited English
proficiency or those who learn more visually. In my answer for item 3, I noted two possible
solutions for the multiplier, which change the answers for each part of the item. By making note
of this, I will ensure that students do not lose points for a different way of thinking that is just as
“correct” as my own.
In order to ensure reliability of the activity, I would grade each paper individually using the same
grading rubric each time (described below in part c). This would ensure that every student is
graded using the same criteria, providing comparable results. If students have special
accommodations that need to be made, I will use a modified grading rubric or provide a modified
activity. This activity is valid because each item was created with a specific learning objective in
mind, so it mirrors exactly what the students should be able to do. Therefore, it provides accurate
feedback to the students and me about what the students know and what they need to work more
Maggie Sharp
EPSY 485
on in order to master the material. The activity is meant to increase self-reflection among
students and to provide feedback to inform my own teaching.
4. Explanation Questions
a. I would collect this assignment and grade it with comments, but I would not count it
for points. The reason I would collect it and grade it would be to provide feedback to
my students about what they have mastered and what they need to work on more. I
think it is beneficial for students to work on an assignment like this a day or two after
they have gone over the concepts in order to determine their comfort with the
material. However, I do not believe that formative assessments like this should be
high stakes, so I would not count the points from the grade. Another benefit of
grading is that I can see where the gaps in my students’ knowledge lie and
appropriately adjust my instruction or go back to address the problems from the
material. I would treat this worksheet as a class activity meant to supplement the
course material and increase understanding and self-reflection of learning among
students.
b. I would conduct this activity as a worksheet to be done individually by students in
class. This activity should not take students more than 15-20 minutes since it is a
review of previous material, so I would have the class do this at the end of the period
after having done group work which pertained to the exponential functions unit. The
group work would be designed to give the students practice with the learning
objectives from the unit including the three listed above. I believe it is most beneficial
for students to do the worksheet on their own because it will reveal their individual
gaps in knowledge more clearly. Additionally, they will already have worked in
groups to talk through any immediate problems they saw before the worksheet is
completed. At the end of class, students would turn in their assignments and I would
grade them with comments to return the next day, which would provide immediate
feedback (explained above). Finally, if I had students with special needs I would
provide appropriate accommodations. For example, I would allow students with IEPs
to complete the worksheet in another room if necessary. Or I would provide the
worksheet in another language if there were English Language Learners in the room.
c. I will use the following individual performance criteria to determine if the students
are meeting the learning objectives (on a scale of 1-3 points):
Item 1 (Learning objective 1):
1- Student can draw general curve but it is inaccurate and key features are not
identified
2- Students can accurately draw the curve on the graph, but they do not identify
key features
3- Students accurately draw curve and key features are accurately identified
Item 2 (Learning objective 2):
1- Students cannot describe or draw more than one of the transformations
accurately
2- Students can describe or draw most of the transformations accurately, but they
leave out important features such as intercepts or asymptotes
Maggie Sharp
EPSY 485
3- Students can describe or draw all of the transformations accurately, noting key
features
Item 3 (Learning objective 3):
1- Students cannot answer any of the questions or they can only answer question
a dealing with identifying the multiplier and percent
2- Students answer all questions and create an equation that is correct or nearly
correct. Students use equation accurately in parts c and d.
3- Students answer all questions accurately and apply the appropriate equation to
the situation given.
Note: In each case, a 1 indicates minimal understanding, a 2 indicates limited understanding, and
a 3 indicates exceptional understanding.
d. In order to determine the performance of my class as a whole, I would average the
points for each item individually. If the class average is a 2.5 or above for any given
item, I will be satisfied that overall, the class has a good mastery of the corresponding
learning objective. If for any item, the average is less than a 2.5, I would want to
revisit that learning objective as a class in order to ensure mastery of the
corresponding learning objective.
e. If the evidence I collected above indicated that the students were below a 2.5 on two
or more items as a class, then when we I hand back the papers the next day, I would
open up the class for questions about the worksheet to begin with. I would attempt to
allow students to field the questions for each other, so if one student asks a question, I
would ask another student or a group of students to answer the question. If no one
feels comfortable with their understanding, I will walk the students through the
question myself. Once students feel they have been given ample time to discuss their
questions from the worksheet, I would summarize the understanding each item sought
to measure by placing the learning objectives on the board and connecting them to
what I was specifically looking for. Once we go over that, I would ask the class for
any general questions. If there is only one learning objective below a 2.5 in the class
average, I would only make time to discuss that item in a similar manner to the
worksheet discussion described above. Finally, all the items demonstrate an
understanding of 2.5 or above, I would move on to the next lesson after passing back
the papers at the beginning of class. In any situation, I would encourage students to
come in to talk to me individually if they have any questions that are not covered
during class time.