### NWEA Math Items (3)

• 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