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Mathematics (09) Practice TestSECOND SAMPLE WEAK RESPONSE FOR OPEN-RESPONSEITEM ASSIGNMENT #1Let t = n – 1E(t) = ae λta = 5005000 = 500e λ(1)5000500 = eλλ = Bn 10 = 2.30E(t) = 500e 2.3ty = mx + bm =5000 – 5002 – 1 = 4500y = 4500x + bb = 500y = 4500x + 500y = mx + b A y = m 2 x2 + bx + cy = 2250x 2 + 500x + c5000 = 2250 + 500 + cc = 2250Q(x) = 2250x 2 + 500x + 2250n = 4E(t) = E(3) = 500e 2.3(3) = 496,137Q(x) = Q(4) = 2250(4) 2 + 500(4) + 2250 = 40,250% error = ⎝⎛E(n) – Q(n) ⎞Q(n) 100% error = ⎛ ⎝% error = 1,133%⎠496,137 – 40,250 ⎞40,250 100⎠84
Mathematics (09) Practice TestANALYSIS FOR SECOND WEAK RESPONSE TO OPEN-RESPONSEITEM ASSIGNMENT #1This is an example of a weak response because it is characterized by the following:Purpose: The candidate demonstrates understanding of the general purpose of the problem but not theknowledge to carry it out correctly. While the candidate demonstrates understanding of the concept of creating anexponential model, the candidate only carries this process out in a memorized standard form. The candidate doesnot show the flexibility to adapt the procedure to the specified form. The candidate does not create a quadraticmodel and attempts to use integral calculus to adapt a linear model. This approach does not work. The candidatedoes carry out the last two charges correctly even though the results are invalidated by errors in the first twocharges.Subject Matter Knowledge: The candidate demonstrates a functional knowledge of techniques for generatingstandard exponential and linear models from a set of data, although an error is made in calculating the y-interceptfor the linear model. The response suggests that the candidate's understanding of mathematical modeling is notdeep enough to apply to a variety of function forms such as exponential functions with various bases. Thecandidate's exponential model does, however, show an understanding of standard exponential modeling. Theattempt at a quadratic model reveals several underlying misconceptions. The candidate is missing the basicpremise that a quadratic function cannot be determined by less than three points. The candidate's misuse ofintegration to generate a quadratic function from a linear function shows serious misconceptions in the field ofintegral calculus. The candidate does, however, correctly demonstrate the use of mathematical models to makepredictions as well as the ability to calculate percent error.Support: The response is clearly and logically laid out. Even though there is no verbal commentary in theresponse, the candidate's reasoning, though flawed, is easy to follow. There is no need to further documentthe candidate's decision-making process. Many of the candidate's decisions are incorrect and based onmisconceptions (e.g., a quadratic model can be derived from only two data points).Rationale: The candidate demonstrates a basic understanding of mathematical modeling. However, the responsesuggests that the candidate's understanding is not deep enough to be transferable to unfamiliar applications. In anattempt to go beyond the candidate's skill level, the candidate tries to use integral calculus. This illuminates notonly the candidate's limitations in mathematical modeling, but also some major misconceptions about the use ofintegral calculus.85
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