1. Item 1:
Specification
Grade IWS Content Clarifications Item Approach & Design
High
School
Model problems of
growth using
exponential
functions.
1. Items should be presented in a real-world
context that is appropriate for algebra 2
students.
2. The item should provide the formula y =
ne^(kt) where n is the initial quantity,t is the
time, and k is the growth constant.
3. In this MC item, model means that the
student can use the information in the context
to solve for k.
1 This item should use a data table with
verifiable data having to do with
population growth.
2 Item should provide a formula students
can use to calculate the growth constant.
3 Other potential topics you could use to
assess understanding ofthis skill are
capitalized interest on a loan or the half-
life of fossils.
SpecificationNumber (SOW_Spec#):29
Key: C Illustration needed: NO Photo Needed: NO
DIRECTIONS
Use the data table below to answer the question.
ASSET (Paste a mock-up of what the asset should look like; providedetails in the Note to Media section of this template)
Year #ZM/m2
1 0.33
2 6.7
3 133
4 2,674
5 53,709
STEM/QUESTION
You will need to use the following formula to answer the question:
● y = nekt where n is the initial quantity, t is the time, and k is the growth constant.
Animal biologists are attempting to determine what the population of zebra mussels will look like after a single
adult zebra mussel was found in Lake Madonna in South Central Wisconsin . They send divers into three
separate locations of Madonna Lake and take the average number of zebra mussels found each year for five
years. Which simplified equation below is a correct expression of the growth constant (k):

2. OPTIONS
A k = ln(53,709)
B k = 0.33(ln(e5))
C k = [ln(162,755)]/5
D k = ln(162,755)
E k = [ln(53,709)]/5
JUSTIFICATIONs
A Incorrect response: Plugging in 53,709 for ‘y’, 0.33 for ‘n’, and 5 for ‘t’ results in the equation 53,709 =
0.33e5k which is simplified by dividing both sides by 0.33 which results in 162,755 = e5k. A student who
chose this answer did not take this step when simplifying this equation.
B Incorrect response: Plugging in 53,709 for ‘y’, 0.33 for ‘n’, and 5 for ‘t’ results in the equation 53,709 =
0.33e5k. A student who chose this as an answer most likely did not plug in 53,709 for ‘y’ before attempting
to simplify.
C Key – correct response: Plugging in 53,709 for ‘y’, 0.33 for ‘n’, and 5 for ‘t’ results in the equation 53,709 =
0.33e5k which is simplified by dividing both sides by 0.33 which results in 162,755 = e5k, taking the natural
log of both sides which results in ln(162,755) = 5k, and dividing both sides of the equation by five in order
to arrive at the correct answer.
D Incorrect response: Plugging in 53,709 for ‘y’, 0.33 for ‘n’, and 5 for ‘t’ results in the equation 53,709 =
0.33e5k which is simplified by dividing both sides by 0.33 which results in 162,755 = e5k, taking the natural
log of both sides which results in ln(162,755) = 5k, and dividing both sides of the equation by five in order
to arrive at the correct answer. A student who chose this answer most likely neglected the last step in
simplifying the equation.
E Plugging in 53,709 for ‘y’, 0.33 for ‘n’, and 5 for ‘t’ results in the equation 53,709 = 0.33e5k which is
simplified by dividing both sides by 0.33 which results in 162,755 = e5k. The student who chose this answer
most likely did not divide both sides by 0.33 before continuing to simplify the expression.
Note to media team: (Please describe any graphics, including changes to sample graphics. Put the source
(URL) and copyright information for the sample graphics here)
URL: http://ats.doit.wisc.edu/biology/ec/pd/t1.htm

3. Credits
Principal Investigator:
Robert Jeanne
Depts. Of Zoology and Entomology
UW-Madison
jeanne@entomology.wisc.edu
Project Manager:
Jan Cheetham
Learning Solutions
UW-Madison
cheetham@wisc.edu
Instructional Designers and Consultants:
Lee Clippard
Learning Solutions
lhclippard@wisc.edu
Alan Wolf
Learning Technology and Distance Education & Center for Biology Education
Les Howles
Learning Solutions
howles@doit.wisc.edu
Project Assistants:
Edna Francisco
Steven Grunder
Sainath Suryanarayanan
Ben Schulte
Olaf Olson
Programmers:
Michelle Glenetski
Learning Solutions
maglenetski@doit.wisc.edu
Bahman Zakeri
Learning Solutions
bahman.zakeri@doit.wisc.edu
Cidney Frietag
Learning Solutions
cfreitag@wisc.edu
Instructors and Instructional Support Staff, Biology 151-152:
Nancy Ruggeri
Jean Heitz
Carlos Peralta
Brian Manske
Catherine Reinitz
Doug Rouse
Carol Lee
Tony Stretton
Dave Abbott
Milo Wittbank
Steve Gammie
Bill Mellon
Ted Golos
Paul Weimer
Tony Bleecker
Eric Triplett
Bob Goodman
Linda Graham
Seth Blair
Donna Fernandez
Rich Viersta
Tom Martin
Tim Allen
John Kirsch
Evelyn Howell
Monica Turner
Stanley Dodson
Kathy Barton
Ken Sytsma
Paul Berry
Tom Sharkey
Edgar Spalding

4. Source documentation (Place references for data here. Please VERIFY ACCURACY and GRADE-APPROPRIATENESS of
content and misconceptions, and validity of data.)
Internal documentation (for INTERNAL use by NWEA ONLY)
Item 2:
Specification
Grade IWS Content Clarifications Item Approach & Design
High
School
Choose sine
functions to model
periodic phenomena
with specified
amplitude, frequency,
and midline.
1. Students are expected to know the
following general form of the sine function:
y = Asin(Bx) + k
*y = k is the midline (horizontal line between
the maximum and minimum values)
*A is the amplitude (distance between
maximum value and midline)
*B is the angular frequency (number of
cycles completed between 0 and 2Pi)
2. Note: Horizontal shifts are not included
here.
3. The "model" used here will be the
graphical representation of the sine function.
4. For this item, k = 0.
5. Do not include a context.
1 Use an image that represents a “periodic
phenomena” such as a wave or a
grandfather clock instead of a standard
coordinate plane graph.
2 Use a contrast of shades of color (if
black, white and gray are all that are
available) in order to make it easy for
the student to distinguish the graph from
the background image.
3 Answer options should be versions of
the correct answer with coefficients
scrambled in different orders.
SpecificationNumber (SOW_Spec#):32
Key: E Illustration needed: YES Photo Needed:NO
DIRECTIONS
The yellow line in the image below represents a cosine function. Use the image below to answer the question.
ASSET (Paste a mock-up of what the asset should look like; providedetails in the Note to Media section of this template)

5. STEM/QUESTION
Which cosine function below is an appropriate model of the graph represented by the yellow line above?
OPTIONS
A Y = 4cos(3x) + 2
B Y = 3cos(4x) + 2
C Y = 2cos(x) + 3
D Y = 2cos(x) + 4
E Y = cos(2x) + 3

6. JUSTIFICATIONs
A Incorrect response: The coefficient of the cosine function (A) represents the amplitude and not the distance
between the x-axis and the midline of the graph. Also, the coefficient of the x-variable (B) represents the
number of cycles completed between pi and 2pi instead of between pi and 3pi. Finally, the constant (k)
represents the distance the graph has been shifted up or down instead of the distance between the x-axis and
the minimum.
B Incorrect response: The coefficient of the cosine function (A) represents the amplitude and not the
maximum value of the graph. Also, the coefficient of the x-variable (B) represents the number of cycles
completed between pi and 2pi instead of the distance between the x-axis and the maximum. Finally, the
constant (k) represents the distance the graph has been shifted up or down instead of the distance between
the x-axis and the minimum.
C The coefficient of the cosine function (A) represents the amplitude and not the distance between the
maximum and the minimum. Also, the coefficient of the x-variable (B) represents the number of cycles
completed between pi and 2pi instead of the amplitude.
D The coefficient of the cosine function (A) represents the amplitude and not the distance between the
maximum and the minimum. Also, the coefficient of the x-variable (B) represents the number of cycles
completed between pi and 2pi and not the amplitude. Finally, the constant (k) represents the distance the
graph has been shifted up or down instead of the distance between the maximum and the minimum.
E Key—Correct Response: Amplitude = 1, Number of Cycles between 0 and 2pi = 2, Graph shifted up by 3.
Note to media team (Please describe any graphics, including changes to sample graphics. Put the source (URL) and copyright
information for the sample graphics here)
I found the original wave image through google images at the following address:
http://www.google.com/imgres?q=waves&hl=en&sa=X&biw=1366&bih=667&tbm=isch&prmd=imvns&tbnid
=VZ22_Up0jF4kSM:&imgrefurl=http://sgfntmj.edu.glogster.com/why-the-waves-have-
whitecaps/&imgurl=http://edu.glogster.com/media/3/10/95/32/10953203.jpg&w=504&h=337&ei=y0RAUMD
KCKXk0QHsz4DIDA&zoom=1&iact=hc&vpx=398&vpy=386&dur=1728&hovh=183&hovw=275&tx=111&t
y=150&sig=116964762425542674494&page=2&tbnh=135&tbnw=202&start=18&ndsp=20&ved=1t:429,r:16,s
:18,i:254
I imported the image into my paint software and drew the yellow cosine wave.
Source documentation (Place references for data here. Please VERIFY ACCURACY and GRADE-APPROPRIATENESS of
content and misconceptions, and validity of data.)
Internal documenation (for INTERNAL use by NWEA ONLY)

7. Item 3:
Specification
Grade IWS Content Clarifications Item Approach & Design
Algebra
II
Understand that
restricting a
trigonometric
function to a domain
on which it is always
increasing or always
decreasing allows its
inverse to be
constructed
1. The focus here is on understanding howto
construct an inverse trigonometric function.
2. The item should not ask the student to
calculate the inverse at a particular value.
3. Restrict trigonometric functions to sin(θ)
and cos(θ).
1 The item must include a graph that does
not have an inverse.
2 Graph can be of a sine function or a
cosine function.
3 Graph must cross the x-axis.
4 Graph must not cross the y-axis.
5 Intervals of coordinate plane must be
labeled.
SpecificationNumber (SOW_Spec#):33
Key: B Illustration needed: YES Photo Needed: NO
DIRECTIONS
Use the diagram below to answer the question.
ASSET (Paste a mock-up of what the asset should look like; providedetails in the Note to Media section of this template)
STEM/QUESTION
Which statement explains why the inverse of the graphed sine function above does not exist without restricting
the domain?

8. OPTIONS
A The graph of the sine wave above does not cross the y-axis.
B The inverse of the graph would not pass the vertical line test.
C The domain of the graphed function is already restricted.
D The graph of the sine wave above crosses the x-axis.
JUSTIFICATIONs
A Incorrect response: Though this may be an observable fact about the graph, it does not serve as an adequate
explanation for why the inverse of the function does not exist without further restricting the domain.
B Key – correct response: The vertical line test is one acceptable way of determining a graph is a
representation of a function. It is reasonable to believe that a high school student taking algebra II would be
able to mentally picture this graph being rotated 90 degrees with a vertical line that intersects the graph at
two different points.
C Incorrect response: Thought this may be a true statement about the graph, it does not adequately explain
why the inverse of this graph does not exist without a further restriction of the domain.
D Incorrect response: Though this may be an observable fact about the graph, it does not serve as an adequate
explanation for why the inverse of the function does not exist without further restricting the domain.
Note to media team (Please describe any graphics, including changes to sample graphics. Put the source (URL) and copyright
information for the sample graphics here)
1 Please create graphs of functions that have a restricted domain in order for Answer Choice to be
considered a viable option to students.
2 Create graphs of functions that restrict domains so that their graphs do not cross the y-axis so that
Answer Choice A is a viable option to students.
3 Create graphs of functions that restrict domains so that their graphs cross the x-axis so that Answer
4 Only include graphs that would not have inverses without restricting the domain of the original function.
Source documentation (Place references for data here. Please VERIFY ACCURACY and GRADE-APPROPRIATENESS of
content and misconceptions, and validity of data.)
● The image of the graph used in this item was created using an online graphing calculator located here:
http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html
● I took a screen shot of the graph I created using the online graphing calculator and used my computer’s
Paint Software to crop this photo and inserted it into this item.
Internal documenation (for INTERNAL use by NWEA ONLY)