| 1.0 Students identify and use the arithmetic properties of subsets of
integers and rational, irrational, and real numbers, including closure properties
for the four basic arithmetic operations where applicable.
|
Algebra I - Sections 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7 |
| 1.1 Students use properties of numbers to demonstrate whether assertions
are true or false.
|
Algebra I - Sections 2.3, 3.1 |
| 2.0 Students understand and use such operations as taking the opposite,
finding the reciprocal, taking a root, and raising to a fractional power. They
understand and use the rules of exponents.
|
Algebra I - Sections 1.1, 1.2, 2.3, 3.1, 9.1, 9.3 |
| 3.0 3.0 Students solve equations and inequalities involving absolute values. |
Algebra I - Sections 7.1, 7.2, 7.3, 7.5 |
| 4.0 Students simplify expressions before solving linear equations and
inequalities in one variable, such as 3(2x-5) + 4(x-2) = 12.
|
Algebra I - Sections 3.1, 3.2, 3.3, 3.4 |
| 5.0 Students solve multistep problems, including word problems, involving
linear equations and linear inequalities in one variable and provide justification
for each step.
|
Algebra I - Sections 3.1, 3.2, 3.3, 7.1, 7.2, 7.3 |
| 6.0 Students graph a linear equation and compute the x- and y- intercepts
(e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear
inequality (e.g., they sketch the region defined by 2x + 6y < 4).
|
Algebra I - Sections 6.1, 6.2, 6.3, 7.4 |
| 7.0 Students verify that a point lies on a line, given an equation of
the line. Students are able to derive linear equations by using the point-slope
formula.
|
Algebra I - Sections 5.3, 5.4, 6.2, 6.3 |
| 8.0 Students understand the concepts of parallel lines and perpendicular
lines and how those slopes are related. Students are able to find the equation of
a line perpendicular to a given line that passes through a given point.
|
Algebra I - Section 6.4 |
| 9.0 Students solve a system of two linear equations in two variables
algebraically and are able to interpret the answer graphically. Students are
able to solve a system of two linear inequalities in two variables and to sketch
the solution sets.
|
Algebra I - Sections 7.3, 7.4, 8.1, 8.2, 8.3 |
| 10.0 Students add, subtract, multiply, and divide monomials and polynomials.
Students solve multistep problems, including word problems, by using these techniques.
|
Algebra I - Sections 9.1, 9.3, 9.4, 9.5 |
| 11.0 Students apply basic factoring techniques to second-and simple
third-degree polynomials. These techniques include finding a common factor for all
terms in a polynomial, recognizing the difference of two squares, and recognizing
perfect squares of binomials.
|
Algebra I - Sections 10.1, 10.2, 10.3 |
| 12.0 Students simplify fractions with polynomials in the numerator and
denominator by factoring both and reducing them to the lowest terms.
|
Algebra I - Section 9.1 |
| 13.0 Students add, subtract, multiply, and divide rational expressions and
functions. Students solve both computationally and conceptually challenging problems
by using these techniques.
|
Algebra I - Sections 2.3, 2.4, 2.5, 2.6, 3.1, 3.2, 3.3 |
| 14.0 Students solve a quadratic equation by factoring or completing the square. |
Algebra I - Section 10.4 |
| 15.0 Students apply algebraic techniques to solve rate problems, work
problems, and percent mixture problems.
|
Algebra I - Sections 4.1, 4.2, 4.3, 4.4, 4.5 |
| 16.0 Students understand the concepts of a relation and a function, determine
whether a given relation defines a function, and give pertinent information about given
relations and functions.
|
Algebra I - Sections 5.2, 5.3, 5.4, 5.5 |
| 17.0 Students determine the domain of independent variables and the range of
dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression.
|
Algebra I - Sections 5.2, 5.3, 5.4, 5.5 |
| 18.0 Students determine whether a relation defined by a graph, a set of
ordered pairs, or a symbolic expression is a function and justify the conclusion.
|
Algebra I - Section 5.5 |
| 19.0 Students know the quadratic formula and are familiar with its proof by
completing the square.
|
Algebra I - Section 11.2 |
| 20.0 Students use the quadratic formula to find the roots of a second-degree
polynomial and to solve quadratic equations.
|
Algebra I - Section 11.2 |
| 21.0 Students graph quadratic functions and know that their roots are the
x- intercepts.
|
Algebra I - Section 11.1 |
| 22.0 Students use the quadratic formula or factoring techniques or both to
determine whether the graph of a quadratic function will intersect the x-axis in zero,
one, or two points.
|
Algebra I - Section 10.4, 11.1 |
| 23.0 Students apply quadratic equations to physical problems, such as the
motion of an object under the force of gravity.
|
Algebra II |
| 24.0 Students use and know simple aspects of a logical argument |
Algebra I - Arguments are based off of definitions and axioms throughout the entire course. |
24.2 Students identify the hypothesis and conclusion in logical deduction.
24.3 Students use counterexamples to show that an assertion is false and recognize
that a single counterexample is sufficient to refute an assertion.
|
Algebra I - Sections 3.1, 3.2, 3.3, 3.4, 11.2 |
| 25.0 Students use properties of the number system to judge the validity of results,
to justify each step of a procedure, and to prove or disprove statements:
|
Algebra I - Chapters 1,2, and 3 |
| 25.1 Students use properties of numbers to construct simple, valid arguments
(direct and indirect) for, or formulate counterexamples to, claimed assertions.
|
Algebra I - Chapters 1,2, and 3 |
| 25.2 Students judge the validity of an argument according to whether the properties
of the real number system and the order of operations have been applied correctly at each step.
|
Algebra I - Chapters 1,2, and 3 |
| 25.3 Given a specific algebraic statement involving linear, quadratic, or absolute
value expressions or equations or inequalities, students determine whether the statement
is true sometimes, always, or never.
|
Algebra I - Present throughout the course. |
State Standards For Algebra II |
| 1.0 Students solve equations and inequalities involving absolute value. |
Algebra II - Section 1.3 |
| 2.0 Students solve systems of linear equations and inequalities (in two or three variables)
by substitution, with graphs, or with matrices.
|
Algebra II - Sections 3.1, 3.2, 3.3, 3.4, 4.5 |
| 3.0 Students are adept at operations on polynomials, including long division. |
Algebra II - Sections 9.1, 9.2, 9.3, 9.4, 9.5, 9.6 |
| 4.0 Students factor polynomials representing the difference of squares, perfect
square trinomials, and the sum and difference of two cubes. |
Algebra II - Sections 9.4, 9.5 |
| 5.0 Students demonstrate knowledge of how real and complex numbers are related
both arithmetically and graphically. In particular, they can plot complex numbers as
points in the plane.
|
Algebra II - Sections 6.4, 6.5 |
| 6.0 Students add, subtract, multiply, and divide complex numbers. |
Algebra II - Sections 6.4, 6.5 |
| 7.0 Students add, subtract, multiply, divide, reduce, and evaluate rational
expressions with monomial and polynomial denominators and simplify complicated rational
expressions, including those with negative exponents in the denominator.
|
Algebra II - Sections 10.2, 10.3 |
| 8.0 Students solve and graph quadratic equations by factoring, completing the square, or
using the quadratic formula. Students apply these techniques in solving word problems.
They also solve quadratic equations in the complex number system.
|
Algebra II - Sections 6.1, 6.2, 6.3, 6.4, 6.5 |
| 9.0 Students demonstrate and explain the effect that changing a coefficient has on the
graph of quadratic functions;
|
Algebra II - Section 5.4, 9.6 |
| 10.1 Students graph quadratic functions and determine the maxima, minima, and zeros of
the function.
|
Algebra II - Section 9.6 |
| 10.2 Students add, subtract, multiply, and divide monomials and polynomials.
Students solve multistep problems, including word problems, by using these techniques.
|
Algebra I - Sections 9.1, 9.3, 9.4, 9.5 |
| 11.0 Students prove simple laws of logarithms. |
Algebra II - Sections 8.1, 8.2, 8.3, 8.4, 8.5 |
| 11.1 Students understand the inverse relationship between exponents and logarithms
and use this relationship to solve problems involving logarithms and exponents.
|
Algebra II - Sections 8.1, 8.2, 8.3, 8.4, 8.5 |
| 11.2 Students judge the validity of an argument according to whether the properties
of real numbers, exponents, and logarithms have been applied correctly at each step.
|
Algebra II - Sections 8.1, 8.2, 8.3, 8.4, 8.5 |
| 12.0 Students know the laws of fractional exponents, understand exponential functions,
and use these functions in problems involving exponential growth and decay.
|
Algebra II - Sections 7.1, 7.2 |
| 13.0 Students use the definition of logarithms to translate between logarithms in any base. |
Algebra II - Sections 8.1, 8.2, 8.3, 8.4, 8.5, 8.6 |
| 14.0 Students understand and use the properties of logarithms to simplify logarithmic
numeric expressions and to identify their approximate values.
|
Algebra II - Sections 8.1, 8.2, 8.3, 8.4, 8.5, 8.6 |
| 15.0 Students determine whether a specific algebraic statement involving rational
expressions, radical expressions, or logarithmic or exponential functions is sometimes true,
always true, or never true.
|
Algebra II - Sections 7.1, 7.2, 7.3, 7.4, 7.5, 8.1, 8.2, 8.3, 8.4, 8.5 |
| 16.0 Students demonstrate and explain how the geometry of the graph of a conic section
(e.g., asymptotes, foci, eccentricity) depends on the coefficients of the quadratic
equation representing it.
|
Algebra II - Sections 11.1, 11.2, 11.3, 11.4, 11.5 |
| 17.0 Given a quadratic equation of the form ax2 + by2 + cx + dy + e = 0, students
can use the method for completing the square to put the equation into standard form
and can recognize whether the graph of the equation is a circle, ellipse, parabola,
or hyperbola. Students can then graph the equation.
|
Algebra II - Section 11.1, 11.2, 11.3, 11.4, 11.5 |
| 18.0 Students use fundamental counting principles to compute combinations and permutations.
|
Algebra II - Section 12.5 |
| 19.0 Students use combinations and permutations to compute probabilities.
|
Algebra II - Section 12.5 |
| 20.0 Students know the binomial theorem and use it to expand binomial
expressions that are raised to positive integer powers.
|
Algebra II - Section 12.4 |
| 21.0 Students apply the method of mathematical induction to prove general
statements about the positive integers.
|
Algebra II - Section 12.6 |
| 22.0 Students find the general term and the sums of arithmetic series
and of both finite and infinite geometric series.
|
Algebra II - Section 12.1, 12.2, 12.3 |
| 23.0 Students derive the summation formulas for arithmetic series and for both
finite and infinite geometric series.
|
Algebra II - Section 12.1, 12.2, 12.3 |
| 24.0 Students solve problems involving functional concepts, such as composition,
defining the inverse function and performing arithmetic operations on functions.
|
Algebra II - Sections 5.1, 5.2, 5.3 |
| 25.0 Students use properties from number systems to justify steps in combining
and simplifying functions.
|
Algebra II - Prevalent throughout the course |
| 1.0 Students demonstrate knowledge of both the formal definition and the
graphical interpretation of limit of values of functions. This knowledge
includes one-sided limits, infinite limits, and limits at infinity. Students
know the definition of convergence and divergence of a function as the domain
variable approaches either a number or infinity.
|
Calculus - Sections 2.1, 2.2, 2.3 |
| 1.1 Students prove and use theorems evaluating the limits of sums, products,
quotients, and composition of functions.
|
Calculus - Sections 2.1, 2.2, 2.3 |
| 1.2 Students use graphical calculators to verify and estimate limits. |
Calculus - Sections 2.1 |
| 1.3 Students prove and use special limits, such as the limits of (sin(x))/x and
(1-cos(x))/x as x tends to 0.
|
Calculus - Sections 2.1, 2.2, 2.3, 3.3 |
| 2.0 Students demonstrate knowledge of both the formal definition and the
graphical interpretation of continuity of a function.
|
Calculus - Section 2.2 |
| 3.0 Students demonstrate an understanding and the application of the intermediate
value theorem and the extreme value theorem.
|
Calculus - Sections 2.2, 4.3 |
| 4.0 Students demonstrate an understanding of the formal definition of the
derivative of a function at a point and the notion of differentiability.
|
Calculus - Section 3.1 |
| 4.1 Students demonstrate an understanding of the derivative of a function
as the slope of the tangent line to the graph of the function.
|
Calculus - Section 3.1 |
| 4.2 Students demonstrate an understanding of the interpretation of the derivative
as an instantaneous rate of change. Students can use derivatives to solve a variety
of problems from physics, chemistry, economics, and so forth that involve the rate
of change of a function.
|
Calculus - Sections 4.1, 4.2, 4.3, 4.4, 4.6 |
| 4.3 Students understand the relation between differentiability and continuity. |
Calculus - Section 3.1 |
| 4.4 Students derive derivative formulas and use them to find the derivatives of algebraic,
trigonometric, inverse trigonometric, exponential, and logarithmic functions.
|
Calculus - Sections 3.2, 3.3, 3.4, 3.5, 3.6, 3.7 |
| 5.0 Students know the chain rule and its proof and applications to the calculation of the
derivative of a variety of composite functions.
|
Calculus - Sections 3.4, 3.5, 3.6, 3.7 |
| 6.0 Students find the derivatives of parametrically defined functions and use implicit
differentiation in a wide variety of problems in physics, chemistry, economics, and so forth.
|
Calculus - Sections 3.5, 4.2, 4.6 |
| 7.0 Students compute derivatives of higher orders. |
Calculus - Section 3.7 |
| 8.0 Students know and can apply Rolle's theorem, the mean value theorem, and L'Hôpital's rule. |
Calculus - Sections 4.4, 4.7 |
| 9.0 Students use differentiation to sketch, by hand, graphs of functions. They can identify
maxima, minima, inflection points, and intervals in which the function is increasing and
decreasing.
|
Calculus - Sections 4.3, 4.4, 4.5 |
| 10.0 Students know Newton's method for approximating the zeros of a function. |
Calculus - Section 4.8 |
| 11.0 Students use differentiation to solve optimization (maximum-minimum problems)
in a variety of pure and applied contexts.
|
Calculus - Section 4.6 |
| 12.0 Students use differentiation to solve related rate problems in a variety
of pure and applied contexts.
|
Calculus - Section 4.2 |
| 13.0 Students know the definition of the definite integral by using Riemann sums.
They use this definition to approximate integrals
|
Calculus - Sections 5.1, 5.2 |
| 14.0 Students apply the definition of the integral to model problems in physics,
economics, and so forth, obtaining results in terms of integrals.
|
Calculus - Section 7.5 |
| 15.0 Students demonstrate knowledge and proof of the fundamental theorem of calculus
and use it to interpret integrals as antiderivatives.
|
Calculus - Sections 5.3, 5.4, 7.1, 7.2, 7.3, 7.4, 7.5, 7.6 |
| 16.0 Students use definite integrals in problems involving area, velocity, acceleration,
volume of a solid, area of a surface of revolution, length of a curve, and work.
|
Calculus - Sections 7.1, 7.2, 7.3, 7.4, 7.5, 7.6 |
| 17.0 Students compute, by hand, the integrals of a wide variety of functions by
using techniques of integration, such as substitution, integration by parts, and
trigonometric substitution. They can also combine these techniques when appropriate.
|
Calculus - Sections 6.1, 6.2, 6.3, 6.4 |
| 18.0 Students know the definitions and properties of inverse trigonometric functions
and the expression of these functions as indefinite integrals.
|
Calculus - Section 5.4 |
| 19.0 Students compute, by hand, the integrals of rational functions by combining
the techniques in standard 17.0 with the algebraic techniques of partial fractions
and completing the square.
|
Calculus - Section 6.4 |
| 20.0 Students compute the integrals of trigonometric functions by using the techniques
noted above.
|
Calculus - Sections 6.1, 6.2, 6.3, 6.4 |
| 21.0 Students understand the algorithms involved in Simpson's rule and Newton's
method. They use calculators or computers or both to approximate integrals numerically.
|
Calculus - Sections 5.1 |
| 22.0 Students understand improper integrals as limits of definite integrals. |
Calculus - Sections 6.5 |
| 23.0 Students demonstrate an understanding of the definitions of convergence
and divergence of sequences and series of real numbers. By using such tests as
the comparison test, ratio test, and alternate series test, they can determine
whether a series converges.
|
Calculus - Sections 8.1, 8.2, 8.3, 8.4, 8.5, 8.6 |
| 24.0 Students understand and can compute the radius (interval) of the convergence
of power series.
|
Calculus - Section 8.6 |
| 25.0 Students differentiate and integrate the terms of a power series in order
to form new series from known ones.
|
Calculus - Section 8.7 |
| 26.0 Students calculate Taylor polynomials and Taylor series of basic functions,
including the remainder term.
|
Calculus - Section 8.8 |
| 27.0 Students know the techniques of solution of selected elementary differential
equations and their applications to a wide variety of situations, including
growth-and-decay problems.
|
Calculus - Section 7.6 |
