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Mathematics (09) Practice TestANALYSIS FOR SECOND STRONG RESPONSE TO OPEN-RESPONSEITEM ASSIGNMENT #2This is an example of a strong response because it is characterized by the following:Purpose: The candidate carries out each charge accurately and completely. The volume of a single ball isderived from the formula for the volume of a sphere by substituting d 2 for r. The volume of a linear foot of therubber stock is derived from the formula for the volume of a cylinder by substituting in the dimensions of the footof stock in inches. The function for the speed of stock feed in feet per second is derived from the quotient of thevolume of three balls and the volume of one foot of rubber stock. The speed function is then graphed accordingto the specifications in the fourth charge.Subject Matter Knowledge: The candidate shows a sophisticated understanding of volume, including the way atwo-to-one ratio between radius and diameter translates to an eight-to-one ratio in the respective volume formulas.The candidate is also able to make the abstract conceptual leap from a one-foot length of stock to a cylinder ofheight equal to 12 inches. The candidate is then able to formulate a response to the third charge by relating theanswers to the first two charges. This demonstrates a high-level understanding of related rates and dimensionalanalysis. Finally, the candidate is able to generate a set of data points from the speed function and use them tocreate a graph. The data points are listed in a manner that shows knowledge of rounding. The points areaccurately located between tick marks showing good number sense. The points are then connected with a smoothcurve showing a qualitative understanding of cubic functions.Support: The response contains no verbal explanations and yet each step is perfectly clear and well supportedmathematically. The presentation is clearly detailed using indentation for secondary substitutions and boxes forfinal results, which makes the entire process clear. The candidate details the development of each equation in avertical progression that is easily followed. By including the units in the first step of the third charge, thecandidate uses dimensional analysis to justify the approach. In the fourth charge, the candidate shows thesubstitution of each value for d and how it is used to generate each value of S(d).Rationale: The solution is clear and succinct while containing enough information to justify each step. Thepresentation is accurate, appropriate, and economical with no extraneous information. The candidatedemonstrates a mastery of all of the content involved in the problem, including derivation of formulas andfunctions, spatial reasoning, related rates, dimensional analysis, algebraic manipulation, and creating graphs.96