May 25th
May 25th
Core-Plus Mathematics 2nd Edition
Course 3
| Unit 1 |
Reasoning and Proof develops student
understanding of formal reasoning in geometric, algebraic, and
statistical
contexts and of basic principles that underlie those reasoning
strategies.
Topics include inductive and deductive reasoning strategies; principles
of logical reasoning—Affirming the
Hypothesis and Chaining Implications; relation among angles formed by
two intersecting lines or by two parallel
lines and a transversal; rules for transforming algebraic expressions
and equations; design of experiments
including the role of randomization, control groups , and blinding;
sampling distribution, randomization test, and
statistical significance. |
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| Unit 2 |
Inequalities and Linear Programming
develops student ability to reason both algebraically and graphically to
solve inequalities in one and two variables, introduces systems of
inequalities in two variables, and develops a
strategy for optimizing a linear function in two variables within a
system of linear constraints on those variables.
Topics include inequalities in one and two variables, number line
graphs, interval notation, systems of linear
inequalities, and linear programming. |
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| Unit 3 |
Similarity and Congruence extends student
understanding of similarity and congruence and their ability to use
those relations to solve problems and to prove geometric assertions with
and without the use of coordinates.
Topics include connections between Law of Cosines, Law of Sines, and
sufficient conditions for similarity and
congruence of triangles, centers of triangles, applications of
similarity and congruence in real-world contexts,
necessary and sufficient conditions for parallelograms, sufficient
conditions for congruence of parallelograms,
and midpoint connector theorems. |
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| Unit 4 |
Samples and Variation extends student
understanding of the measurement of variability, develops student
ability
to use the normal distribution as a model of variation, introduces
students to the binomial distribution and its use
in decision making, and introduces students to the probability and
statistical inference involved in control charts
used in industry for statistical process control.
Topics include normal distribution, standardized scores, binomial
distributions (shape, expected value, standard
deviation), normal approximation to a binomial distribution, odds,
statistical process control, control charts, and
the Central Limit Theorem. |
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| Unit 5 |
Polynomial and Rational Functions extends
student ability to represent and draw inferences about polynomial
and rational functions using symbolic expressions and manipulations.
Topics include definition and properties of polynomials, operations on
polynomials; completing the square, proof
of the quadratic formula , solving quadratic equations (including complex
number solutions), vertex form of
quadratic functions; definition and properties of rational functions,
operations on rational expressions. |
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| Unit 6 |
Circles and Circular Functions develops
student understanding of relationships among special lines, segments,
and angles in circles and the ability to use properties of circles to
solve problems; develops student understanding
of circular functions and the ability to use these functions to model
periodic change; and extends student ability to
reason deductively in geometric settings.
Topics include properties of chords, tangent lines, and central and
inscribed angles of circles; linear and angular
velocity; radian measure of angles; and circular functions as models of
periodic change. |
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| Unit 7 |
Recursion and Iteration extends student
ability to represent, analyze, and solve problems in situations
involving
sequential and recursive change.
Topics include iteration and recursion as tools to model and analyze
sequential change in real-world contexts,
including compound interest and population growth; arithmetic,
geometric, and other sequences; arithmetic and
geometric series; finite differences; linear and nonlinear recurrence
relations; and function iteration, including
graphical iteration and fixed points. |
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| Unit 8 |
Inverse Functions develops student
understanding of inverses of functions with a focus on 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; and inverse
trigonometric functions and their applications to
solving trigonometric equations. |
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Course 4—Preparation for Calculus
Part A
| Unit 1 |
Families of Functions extends student
understanding of linear, exponential, quadratic, power, and
trigonometric
functions to model data patterns whose graphs are transformations of
basic patterns; and develops understanding
of operations on functions useful in representing and reasoning about
quantitative relationships.
Topics include linear, exponential, quadratic, power, and trigonometric
functions; data modeling; translation,
reflection, and stretching of graphs; and addition, subtraction,
multiplication, division, and composition of
functions. |
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| Unit 2 |
Vectors and 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, dot product of two
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 |
Algebraic Functions and Equations reviews
and extends student understanding of properties of polynomial and
rational functions and skills in manipulating algebraic expressions and
solving polynomial and rational equations,
and develops student understanding of complex number representations and
operations.
Topics include polynomials, polynomial division, factor and remainder
theorems, operations on complex
numbers, representation of complex numbers as vectors, solution of
polynomial equations, rational function
graphs and asymptotes, and solution of rational equations and equations
involving radical expressions . |
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| Unit 4 |
Trigonometric Functions and Equations
extends student understanding of, and ability to reason with,
trigonometric functions to prove or disprove trigonometric identities
and to solve trigonometric equations; to
geometrically represent complex numbers and complex number operations
and to find powers and roots of
complex numbers expressed in trigonometric form.
Topics include fundamental trigonometric identities, sum and difference
identities, double-angle identities;
solving trigonometric equations and expression of periodic solutions;
secant, cosecant, and cotangent functions;
absolute value and trigonometric form of complex numbers, De Moivre’s
Theorem, and roots of complex
numbers. |
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Part B
| Unit 5 |
Exponential Functions, Logarithms, and Data
Modeling extends student understanding of exponential and
logarithmic functions to the case of natural exponential and logarithmic
functions, solution of exponential growth
and decay problems, and use of logarithms for linearization and modeling
of data patterns.
Topics include exponential functions with rules in the form f(x) = Aekx,
natural logarithm function, linearizing
bivariate data and fitting models using log and log-log transformations. |
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| Unit 6 |
Surfaces and Cross Sections 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.
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 loci of
points in a plane; three-dimensional rectangular coordinate system ;
sketching surfaces using traces, intercepts and
cross sections derived from algebraically-defined surfaces; and surfaces
of revolution and cylindrical surfaces. |
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| Unit 7 |
Concepts of Calculus develops student
understanding of fundamental calculus ideas through explorations in a
variety of applied problem contexts and their representations in
function tables and graphs.
Topics include instantaneous rates of change, linear approximation, area
under a curve, and applications to
problems in physics, business, and other disciplines. |
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| Unit 8 |
Counting Methods and Induction extends
student ability to count systematically and solve enumeration
problems, and develops understanding of, and ability to write, proofs by
mathematical induction.
Topics include systematic listing and counting, counting trees, the
Multiplication Principle of Counting, Addition
Principle of Counting, combinations, permutations, selections with
repetition; the binomial theorem, Pascal’s
triangle, combinatorial reasoning ; the general multiplication rule for
probability; and the Principle of
Mathematical Induction. |
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