Primary Text
Supplementary Materials
Technology
Assessment
·
Tests: Each 9 week period includes 3 tests which
count as 60% of the students’ average. Each test is comprised of AP-style
problems (multiple choice and free response). Some tests require the use of a
calculator; others exclude it.
·
Daily Grades: Homework is required and graded
·
Quizzes: Quizzes are given once or twice per
week and typically spotlight a specific base skill required for the AP Test
(e.g. calculator competencies, basic integral/derivative rules)
·
Semester Exams: These cumulative examinations
count 20% towards a student’s semester grade. They are essentially mini-AP
tests consisting of multiple choice and free response questions, timed in
accordance with standard AP allowances.
·
Projects: During the fall semester, students
engage in (both individually and as a group) in various minor projects.
§ Minor
Project Example: Optimizing the shape of a vegetable can
§ Minor
Project Example: Modeling the disappearance of a tootsie roll pop with related
rates
Course Prospectus
This year-long course is equivalent to a first college course in calculus. Functions, Limits, Continuity, Derivatives, Integrals, and Differential Equations will be covered. The advanced placement curriculum is designed to mirror the progressive approach to mathematics now employed by universities. Unlike traditional math curricula in which answer-focused algorithmic mechanisms are emphasized, the College Board course is designed to provide a conceptually deep understanding of the included concepts, in addition to the traditional algorithmic proficiencies. As a result, answers per se are deemphasized; the cognitive approach used to achieve those answers is emphasized. All topics will be examined in a rich context; each major theme is approached numerically, graphically, analytically, and verbally.
In accordance with the policies of the College Board this course will provide students not only with the generic algorithms utilized in calculus, but empower them to communicate mathematics both orally and in well-written sentences. Students will be able to explain solutions to problems, model physical situations with functions or differential equations, use technology to help solve problems, experiment, interpret results, verify conclusions, and determine the reasonableness of solutions including sign, size, relative accuracy, and units of measurement.
Communications
The ability to communicate mechanisms of mathematical reasoning efficiently via writing is the most important skill students will develop in this course. Written problem solutions must be written up neatly and completely in pencil. Students will be required to argue theoretically from the evidence which they have been given. Students must be able to communicate clearly (with words, numbers, mathematical symbols, and graphs) their understanding of both a particular problem and the answers they find. Please note, “The calculator did it” is not a viable method of explaining your understanding. Failure to communicate mathematics neatly and in pencil will result in point reductions.
Problem Solving
Problem solving is the essence of mathematics. Throughout this calculus course students are expected to problem solve. Student will be introduced to a problem solving heuristic at the beginning of the course and for each topic will be expected to complete a problem solving exercise.
The importance of problem solving in this course is as the unifier of the all the course aspects described above and below. The students will learn all the specific topics (with accompanying theorems and procedures) listed below. In the problem solving exercises they will be challenged to apply their learning into a new context. In course of problem solving students will begin to work in groups, in which students can enhance their communication skills. Not only are students responsible for communicating via writing the problem approach, but they will also have to negotiate verbally, exchange suggestions for approaching a difficult problem, and most importantly verbalize their conceptual understanding.
As the second semester begins, students are challenged to heighten the level of their problem solving as they become accustomed to the AP-type problems. One such activity is based on the 2002’s Amusement Park Free Response Item. We use Play-Doh® and cut-out shapes to help student cognitively construct the notion of volumes with fixed cross sections. Finally the culminating project is a Clue®-inspired calculus murder mystery which I have designed. The students work in groups of two to determine who killed our victim by applying differential equations for Newton’s laws of motion and cooling, and the decay of poison. In part to its open-ended nature, it is the hardest problem which they will encounter, but it is also a fine approximation of an authentic “real-world” problem solving situation.
First
Semester Outline:
Topics |
Book
Sections |
Activities |
I.
Pre-Calculus Materials [2 weeks] A. Four ways to
represent a function: B. Introduction to
Program Solving (Polya’s 4 steps) C. Parent Functions
Review D. Parametric
Functions E.
Geometric Function Behavior a.
Extreme Value Theorem b. Intermediate
Value Theorem c.
Fundamental Theorem of Algebra F.
Function Growth a.
Polynomial vs. Exponential vs. Logarithmic |
1.1-1.7 |
First Problem Solving |
II.
Limits [4 weeks] A. Asymptotic
Behavior a.
Horizontal b. Vertical B. Finding limits a.
Numerically b. Graphically C. Limit Notation &
Laws D. Finding Limits
algebraically E.
Continuity & One Sided Limits F.
Infinite Limits |
2.2-2.5 |
Problem Solving (179) |
III.
Conceptual Derivatives [4 weeks] A. Secant approach
& Average Rate of Change B. Local Linearity C. Tangent Approach
& Instantaneous Rate of Change D. The Definition of
the Derivative as a Numerical limit (of a difference quotient) a.
Epsilon – Delta Definition E.
Continuity & Graphical Interpretation of the
Derivative a.
Does Differentiability Imply Continuity? b. Does Continuity
Imply Differentiability? |
2.1, 2.3-2.9 |
Finding the derivative
graphically Finding the derivative numerically |
IV.
Algebraic Derivatives [4 weeks] A. Polynomial
Derivatives a.
Power Rule b. The derivative as
a linear operator (Scalar & Addition Rules) B. The Product &
Quotient Rules C. Derivatives of
Exponential, Logarithmic, Trigonometric, and Inverse Functions D. The Chain Rule E.
Implicit Derivatives |
3.1-3.8 |
Minor Project: Taylor Polynomials (254) |
V.
Derivative Applications [2 weeks] A. Connecting the
Graphical and Algebraic a.
Increasing vs. Decreasing (First Derivative Test) b. Concavity (Second
Derivative Test) B. What does f tells
us about f’ C. Related Rates D. Minimums &
Maximums (Optimization) E.
Geometric Consequences of Mean Value Theorem F.
Indeterminate Forms and L’Hopital’s Rule |
4.1-4.6 |
Minor Project: Optimizing the Shape of a Can (316) Problem Solving (339) |
Second
Semester Outline:
Topics |
Book
Sections |
Activities & Assignments |
VI.
Basic Integrals [3 weeks]
i.
Biological, Physical, Economic
|
4.9-5.3, 6.4 |
Problem Solving (437) |
VII.
Integrals [4 weeks]
|
5.4-5.6, 5.10 |
Minor Project: Integrals at the Amusement Park |
VIII.
Differential Equations [3 weeks]
|
7.1-7.4 |
Calculus Murder Mystery |
IX.
Integral Applications [3 weeks]
|
6.1-6.2 |
Play-Doh Discovery Problem Solving (496) |
X.
Test Review |
Assorted |
Various Problem Solving |
XI.
After the AP Test |
e: The story of a number |
Various Problem Solving |