Khoon Yu Tan Math Teacher John H Reagan High School Houston Independent School District Dr. Wilbert Wilhelm Barnes Professor Industrial and Systems Engineering Department Texas A&M University

Research overview Element of research for classroom project College Board AP Calculus learning objectives Day one of project Days two and three of project Day four of project Sample pre-test and post-test Acknowledgements

Research overview Element of research for classroom project College Board AP Calculus learning objectives Day one of project Days two and three of project Day four of project Sample pre-test and post-test Acknowledgements

A particular difficulty in deciding capacity configurations while maximizing total revenue excess is uncertainty in patient demand To allow the model to deal with patient uncertainty, expected excess capacity and expected excess demand functions are introduced and analyzed The expected excess demand, E[u] and expected excess capacity, E[o] functions are as follow respectively: where K represents capacity, M mean of patient demand, standard deviation of patient demand, and cummulative (probability) density function of patient demand. Credit: Xue (Lulu) Han

Research overview Element of research for classroom project College Board AP Calculus learning objectives Day one of project Days two and three of project Day four of project Sample pre-test and post-test Acknowledgements

The key element are the expected excess demand and expected excess capacity functions: Students use analytical and graphical tools of Calculus and Pre-Calculus to study the functions The study of the functions helps improve the model that solves the healthcare facility configuration problem, which in turn helps to improve capacity-setting decisions, increasing cost effectiveness Credit: Xue (Lulu) Han

Research overview Element of research for classroom project College Board AP Calculus learning objectives Day one of project Days two and three of project Day four of project Sample pre-test and post-test Acknowledgements

Asymptotes credit: Wolfram MathWorldDerivatives credit: I.Functions, Graphs, and Limits a) Analysis of graphs b) Asymptotic and unbounded behavior: understanding asymptotes in terms of graphical behavior II. Derivatives a) Concept of the derivative: derivative presented graphically, numerically, and analytically b) Derivative at a point: i) slope of a curve at a point ii) tangent line to a curve at a point and local linear approximation c) Derivative as a function: relationship between the increasing and decreasing behavior of f and the sign of f’ d) Second derivatives: relationship between the concavity of f and the sign of f’’ e) Applications of derivatives: analysis of curves, including the notions of monotonicity and concavity

Research overview Element of research for classroom project College Board AP Calculus learning objectives Day one of project Days two and three of project Day four of project Sample pre-test and post-test Acknowledgements

E3 experience: industrial engineering, healthcare data, mathematical model Students inquire pre-day one about their or their parents’ healthcare spending: insurance, out-of-pocket cost Poster, pictures, video, authentic data from students Engage credit: credit: ENGAGE Engineering fields and what engineers do in each Matching-cards game using pictures of engineering work from students’ everyday life & names of engineering fields Think-pair-share and class sharing Shortest flight time: discuss variables e.g. distance, route, time, speed, airport location EXPLORE

Research overview Element of research for classroom project College Board AP Calculus learning objectives Day one of project Days two and three of project Day four of project Sample pre-test and post-test Acknowledgements

Students are to investigate the convexity of the expected excess demand & capacity functions Students are to linearly approximate the functions within a certain percentage error Students follow the engineering design process Explain credit: EXPLAIN Design Process credit: Project Lead The Way

Students are grouped in threes Groups are differentiated by tasks: investigate convexity with respect to different variables Each group builds approximation models after selecting its approach The models differ in terms of approximation error, complexity (piecewise), computational cost (time) Each group selects its “best” model and presents its findings to the class Elaborate credit: ELABORATE Design Process credit: Project Lead The Way

Research overview Element of research for classroom project College Board AP Calculus learning objectives Day one of project Days two and three of project Day four of project Sample pre-test and post-test Acknowledgements

When a group presents, other groups and teacher evaluate based on a rubric Within each group, members anonymously evaluate each other’s contribution Rubric clarifies assignment of points for different steps of the design process, reasoning, conclusion, and presentation Post-test Evaluate credit: EVALUATE

Research overview Element of research for classroom project College Board AP Calculus learning objectives Day one of project Days two and three of project Day four of project Sample pre-test and post-test Acknowledgements

Why is excess demand an important factor in deciding where to open a healthcare facility, or to expand, contract or close an existing facility? I.Excess demand is costly II.Excess demand influences the decision to open a new facility or expand an existing one III.Excess demand causes loss of business (patients) IV.Some patients may be unhappy a)I and II onlyb) II and III only c) I, II, and III onlyd) all of them QUALITATIVE Differentiate the function with respect to capacity level assuming other variables are held constant? *Four multiple-choice responses but correct response is QUANTITATIVE

Research overview Element of research for classroom project College Board AP Calculus learning objectives Day one of project Days two and three of project Day four of project Sample pre-test and post-test Acknowledgements

Texas A&M University E3 Program Dwight Look College of Engineering National Science Foundation Nuclear Power Institute Chevron Dr. Wilbert Wilhelm, faculty adviser Xue (Lulu) Han, Ph.D. candidate Amy Brown, RET participant David Carmona & Brittany Tarin, REU participants