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- 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)