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CPMP
Course 4 1st Edition Units
The mathematical content and sequence of units in Course 4 allows considerable flexibility in tailoring a course to best prepare students for various undergraduate programs. For students intending to pursue programs in the mathematical, physical, and biological sciences or engineering, we recommend the following sequence of units:
Unit 1 - Unit 2 - Unit 3 - Unit 4 - Unit 6 - Unit 7 - Unit 8 - Unit
5, 9, or 10
For students intending to pursue programs in the social, management, and some of the health sciences or humanities, the following sequence of units is recommended:
Unit 1 - Unit 2 (reduced) - Unit 3 - Unit 4 (reduced) - Unit 5 - Unit
9 - Unit 10
Depending on time available, additional units of study can be selected based on student performance and interests.
The CPMP four-year curriculum aligns well with the recommended preparation
for calculus topics in the College Board AP Calculus Guide.
| Unit
1 (sample
material) |
- Rates
of Change
- Develops student understanding of the fundamental concepts underlying calculus and their applications.
|
- Topics
include:
- Average and instantaneous rates of change, derivative at a point and derivative functions, accumulation of continuously varying quantities by estimation, the definite integral, and intuitive development of the fundamental theorem of calculus.
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| Unit
2 (sample
material) |
- Modeling
Motion
- Develops student understanding of two-dimensional vectors and their use in modeling linear, circular, and other nonlinear motion.
|
- Topics
include:
- Concept of vector as a mathematical object used to model situations defined by magnitude and direction; equality of vectors, scalar multiples, opposite vectors, sum and difference vectors, position vectors and coordinates; and parametric equations for motion along a line and for motion of projectiles and objects in circular and elliptical orbits.
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| Unit
3 (sample
material) |
- Logarithmic
Functions and Data Models
- Develops student understanding of logarithmic functions and their use in modeling and analyzing problem situations and data patterns.
|
- Topics
include:
- Inverses of functions; logarithmic functions and their relation to exponential functions, properties of logarithms, equation solving with logarithms; logarithmic scales and re-expression, linearizing data, and fitting models using log and log-log transformations.
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| Unit
4 (sample
material) |
- Counting
Models
- Extends student ability to count systematically and solve enumeration problems, and develops understanding of, and ability to do, proof by mathematical induction.
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- Topics
include:
- Systematic
counting, the Multiplication Principle of Counting, combinations,
permutations; the Binomial Theorem, PascalØs triangle, combinatorial
reasoning; the General Multiplication Rule for Probability;
and the Principle of Mathematical Induction.
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| Unit
5 (sample
material) |
- Binomial
Distributions and Statistical Inference
- Extends student understanding of the binomial distribution, including its exact construction and how the normal approximation to the binomial distribution is used in statistical inference to test a single proportion and to compare two treatments in an experiment.
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- Topics
include:
- Binomial probability formula; shape, mean, and standard deviation of a binomial distribution; normal approximation to a binomial distribution; hypothesis test for a proportion; design of an experiment; randomization test; and hypothesis test for the difference of two proportions.
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| Unit
6 (sample
material) |
- Polynomial
and Rational Functions
- Extends student ability to use polynomial and rational functions to represent and solve problems from real-world situations while focusing on symbolic and graphical patterns.
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- Topics
include:
- Factored and expanded symbolic forms, computational complexity, connections between symbolic and graphical representations, multiplicity of zeroes, end behavior; Factor Theorem, Remainder Theorem, complex numbers and their use in the solution of polynomial equations, Fundamental Theorem of Algebra; equivalent forms of rational expressions; horizontal, vertical, and oblique asymptotes; and optimization.
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| Unit
7 (sample
material) |
- Functions
and Symbolic Reasoning
- Extends student ability to manipulate symbolic representations of exponential, logarithmic, and trigonometric functions; to solve exponential and logarithmic equations; to prove or disprove that two trigonometric expressions are identical and to solve trigonometric equations; to reason with complex numbers and complex number operations using geometric representations and to find roots of complex numbers.
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- Topics
include:
- Equivalent forms of exponential expressions, definition of e and natural logarithms, solving equations using logarithms and solving logarithmic equations; the tangent, cotangent, secant, and cosecant functions; fundamental trigonometric identities, sum and difference identities, double-angle identities; solving trigonometric equations and expression of periodic solutions; rectangular and polar representations of complex numbers, absolute value, DeMoivre's Theorem, and the roots of a complex number.
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| Unit
8 (sample
material) |
- Space
Geometry
- Extends student ability to visualize and represent three-dimensional shapes using contours, cross sections, and reliefs and to visualize and represent surfaces and conic sections defined by algebraic equations.
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- Topics
include:
- Using contours to represent three-dimensional surfaces and developing contour maps from data; sketching surfaces from sets of cross sections; conics as planar sections of right circular cones and as locus of points in a plane; three-dimensional rectangular coordinate system; sketching surfaces using traces, intercepts and cross sections derived from algebraically-defined surfaces; surfaces of revolution and cylindrical surfaces.
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| Unit
9 (sample
material) |
- Informatics
- Develops student understanding of the mathematics of information processing, focusing on the basic issues of access, security, and accuracy.
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- Topics
include:
- Set theory; modular arithmetic; symmetric-key and public-key cryptosystems; error-detecting, including bar codes and check digits.
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| Unit
10 (sample
material) |
- Problem
Solving, Algorithms, and Spreadsheets
- Develops student understanding and skill in use of standard spreadsheet operations for mathematical problems, while at the same time reviewing and extending many of the basic topics in Courses 1-3.
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- Topics
include:
- Mathematics of finance, modeling population growth, apportionment of power in representative governments, sequences and series, and numerical solution of equations.
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| Scope
and Sequence (pdf
- 548Kb) of topics across Courses 1-4 |
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