ADVANCED CALCULUS

Students in SVGS Advanced Calculus wrestled with extending what they learned in calculus last year to three-dimensional space. Along the way, they mastered the use of software tools which (in addition to being powerful problem solving assistants) help us visualize the diﬃcult new concepts of vector calculus. Here’s a link to a webpage containing student – created animations which helped us see what TNB frames and osculating circles are and how they relate to trajectories in 3D space: http://www.svgs.k12.va.us/web/math/ tnb computer creates all the frames in those between the ﬁrst frame and the last and the students do not have to draw all the frames. Students learned how to do program in Action Script to control the buttons and thus the action of the animation. Students created a tutorial using Flash skills and their Photoshop work. Here is an example of a student’s work in Photoshop.

AP CALCULUS

Our AP Calculus BC class began the year with a study of limits and continuity. We then explored the tangent line problem which led into a unit on diﬀerential calculus. We applied diﬀerentiation techniques to a wide variety of functions represented analytically, graphically, and numerically. We then turned our attention to integral calculus where we studied anti-derivatives and Riemann sums as a lead-in to the Fundamental Theorems of Calculus. We investigate diﬀerential equations, inﬁnite series, and applications of calculus to ﬁnd areas and volumes, applied the theorems extensively to physical situations and expanded our repertoire of integration and additional types of functions. AP Calculus students continued to learn new techniques of integration in the second semester of the course. They used calculus to find areas and volumes and they applied calculus to parametric and polar functions. In the final unit of the year, they studied infinite sequences and series and used calculus to approximate complex functions with polynomials. An intensive review at the end of the course included a full-length practice test as students polished their skills for the AP Calculus BC exam.

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AP Statistics students learned to interpret information from graphical and numerical displays and summaries, to collect data according to a well-developed plan. They also studied probability as a tool used for anticipating what the distribution of data should look like under a given model. AP Statistics students learn how to use graphing calculators and computer outputs to enhance the development of statistical understanding through exploring and analyzing data, assessing models, and performing simulations. Through electronic presentations, projects, and models assessed as labs or formal assessments, students were to demonstrate their mastery of content with a variety of products and processes. The next semester will be dedicated to inference and drawing conclusions with conﬁdence. The second semester focus in AP Statistics was on Inference. Students used the theory, methods, and practice to form judgments about the parameters of a population, starting from a drawn random sample. They estimated with confidence, applied tests of significance, and compared two sample means or two proportions. The last material we covered was Inference for two-way tables and inference for Linear Regression. Students were given a practice AP Exam as the class exam, and they took the AP Statistics Exam on May 9^{th}, 2014.

In SVGS Calculus, we worked hard this semester extending ideas from algebra, geometry, and trig into a world of change. Students are learning to use software tools like Maple to help visualize tough new concepts, like the limit deﬁnition of the deﬁnite integral illustrated by the student lab shown to the right. You are welcome to track our day-to-day progress by visiting the class webpage on Moodle. Just click “Login as a guest.” The site has links to online homework, labs, classwork handouts, lecture notes, and test reviews.

Each year, on or near March 14 (3.14), SVGS calculus holds its annual International Pi Day (and Albert Einstein’s birthday) celebration. Part of that celebration is the Digits of Pi competition, in which students in each section compete for a $Pi prize by reciting as many digits of pi as they can remember. This year’s winners were:

Block 1: Seth Jones

Block 2: Ethan Whitehead

Block 3: Seth Wood

Block 4: Aaron White

Grand Champ for 2014: Ethan Whitehead with 211 digits of Pi ! This makes Ethan not only the grand champ for this year, but the Mega-Champion of the SVGS Pi Day Hall of Fame (upsetting Emily Hewitt, who reigned since 2011 with 163 digits).

DISCRETE MATH

In the 1^{st }semester we have studied three major topics: mathematical logic and set theory – foundations of discrete mathematics, number theory – important algorithms and cryptography, and mathematical induction – one of the toughest topics in DM. Math induction is a way to see connections between one case and another and to knock down all the “domino pieces” at once. It is a rigorous logical reasoning method with a great variety of applications in number theory, mathematical logic in computer science and many math issues related to well-founded structures. In the 2^{nd }semester, we will start to learn advanced counting techniques, relations, graph-tree theory, logic gates and mathematical modeling. In the 2nd semester we studied four major topics: relations, graphs, trees and logic gates. We studied how computers represent a relation as two sets and as matrices. For the first time students learned some basics of Microsoft-Access and used it to retrieve data from and produce reports based on a relational database. Euler’s Graphs theory was introduced and applied to social network (below) and communication systems. Trees were studied as a counting tool and spanning trees were focused on for application for diagnostic assessments of communication systems. Finally Boolean algebra was introduced with logic gates for building basic logic circuits like light-systems and adders.

In the ﬁrst semester of Precalculus, students deepened their understanding of what functions are, how they work, how they relate to their graphs and broadened their exposure to diﬀerent kinds of math with new functions, operations, algorithms. They investigated and identiﬁed the characteristics of polynomial and rational, exponential and logarithmic functions and used these characteristics to sketch the graphs of the functions. Precalculus students investigated asymptotic and unbounded behavior in functions, found compositions of functions and inverses of functions, expanded binomials through the use of the Binomial Theorem, and Pascal’s triangle.

Eﬀorts were made to incorporate science, especially physics, concepts into our studies. Three new labs / projects were designed for the purposes. We also started to learn Maple for graphing and demonstrating the properties of those functions and their transformations.

In the last weeks of ﬁrst semester students learned about the Unit Circle and trigonometric functions. We have studied analytic trigonometry and applications, vectors and polar functions, conic sections and sequence-series. Particularly in the first three topics, we emphasized the connection between and combination of those concepts and math skills. Students conducted a project about conic sections functions in real world application. Using mathematic concepts, equations and evaluation of parameters, students conducted research explaining and illustrating how cooling towers are built with stability, how the whispering gallery effect works and what causes a sonic boom (see picture) is caused. In addition, students learned about polar coordinates, analytic geometry and mathematical induction. The last part of second semester was dedicated to SAT Subject Test Level 2 preparation.