Download Algebra I and II - Saddlebrook Preparatory School

Mathematics Common Core State Standards
High School Algebra I and II
COMMON CORE CLUSTER
STANDARDS
College and Career Readiness Cluster
Standards for Mathematics
1. Interpret the structure of
expressions.
COMMON CORE STATE STANDARD
WHAT DOES THIS STANDARD MEAN THAT A STUDENT
MUST KNOW AND BE ABLE TO DO?
Algebra I and II
Linear, quadratic, and exponential expressions
are the focus in Algebra I and integer
exponents extended to rational exponents –
square and cubed roots. Algebra II extends to
polynomial and rational expressions.
Seeing Structure in Expressions
1. Interpret expressions that represent a quantity Students identify the different parts of the expression and explain
in terms of its context.
their meaning within the context of the problem.
a. Interpret parts of an expression, such as
terms, factors, and coefficients.
Students decompose expressions and make sense of the multiple
b. Interpret complicated expressions by
factors and terms by explaining the meaning of the individual parts.
viewing one or more of their parts as a
single entity. For example, interpret P(1 Students classify polynomials by degree, term, even, or odd.
+r)ᴺ as the product of P and a factor not
depending on P.
Students factor polynomials of degree two and higher.
Students identify parts of exponential decay and growth.
Students use the appropriate models for simple compound interest
and continuously compounded interest.
2. Use the structure of an expression to identify
ways to rewrite it.
Students rewrite algebraic expressions in different equivalent forms
such as factoring or combining like terms.
Students rewrite between the three different forms of linear
equations.
Students rewrite a quadratic in factored form.
Students rewrite an inverse variation function in terms of its
constant.
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Students add, subtract, multiply, and divide to write a simpler form
of a rational expression.
2. Write expressions in equivalent
forms to solve problems.
3. Choose and produce an equivalent form of an
expression to reveal and explain properties of
the quantity represented by the expression.
a. Factor a quadratic expression to reveal the
zeroes of the function it defines.
b. Complete the square in a quadratic
expression to reveal the maximum or
minimum value of the function it defines.
c. Use the properties of exponents to
transform expressions for exponential
functions.
Students write expressions in equivalent forms by factoring to find
the zeroes of a quadratic function and explain the meaning of the
zeroes.
4. Derive the formula for the sum of a finite
geometric series (when the common ratio is
not 1), and use the formula to solve
problems.
Students use the formula to solve real world problems such as
calculating the height of a tree after n years given the initial height
of the tree and the rate the tree grows each year.
Students write expressions in equivalent forms by completing the
square to convey the vertex form to find the maximum or minimum
value of a quadratic function, and explain the meaning of the vertex.
Students identify parts of exponential decay and growth.
Students use properties of exponents (power of a power, product of
powers, power of a product and rational exponents) to write
equivalent form of an exponential function to explain the
approximate rate of growth or decay.
Students use the geometric series formula to solve a problem.
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Mathematics Common Core State Standards
High School Algebra I and II
COMMON CORE CLUSTER
STANDARDS
College and Career Readiness Cluster
Standards for Mathematics
1. Perform arithmetic operations on
polynomials.
2. Understand the relationship between
zeroes and factors of polynomials to
perform multi-digit arithmetic.
COMMON CORE STATE STANDARD
WHAT DOES THIS STANDARD MEAN THAT A STUDENT
MUST KNOW AND BE ABLE TO DO?
Algebra I and II
Linear and quadratic polynomial expressions
are the expectation in Algebra 1 and beyond
quadratic polynomial expressions is the
expectation for Algebra II.
Arithmetic with Polynomials and Rational Expressions
1. Understand that polynomials form a system
Students understand the definition of a polynomial.
analogous to the integers, namely, they are
closed under the operations of addition,
Students combine like terms and understand closure.
subtraction, and multiplication; add, subtract,
and multiply polynomials.
Students add, subtract, and multiply polynomials and understand
how closure applies under these operations.
Students apply the Remainder Theorem.
2. Know and apply the Remainder Theorem: For
a polynomial p(x) and a number a, the
remainder on division by x – a is p(a), so p(a)
= 0 if and only if (x – a is a factor of p(x).
3. Identify zeroes of polynomials when suitable
factorization are available, and use the zeroes
to construct a rough graph of the function
defined by the polynomial.
Students understand that a is a root of a polynomial function if and
only if x – a is a factor of the function.
Students use synthetic division to find the remainder or a
polynomial.
Students find the zeroes of a polynomial when the polynomial is
factored.
Students use the zeroes of a function to sketch a graph of the
function.
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3. Use polynomial identities to solve
problems.
4. Prove polynomial identities and use them to
describe numerical relationships. For example,
the polynomial identity (x² + y²)² = (x² - y²)² +
(2xy)² can be used to generate Pythagorean
triples.
Students calculate polynomial identities that include but are not
limited to the product of the sum and difference of two terms, the
difference of two squares, the sum and difference of two cubes, and
the square of a binomial.
Students prove polynomial identities by showing steps and
providing reasons.
5. (+) Know and apply the Binomial Theorem for
the expansion of (x + y)ᴺ in powers of x and y
for a positive integer n, where x and y are any
numbers, with coefficients determined for
example by Pascal’s Triangle.
4. Rewrite rational expressions.
6. Rewrite simple rational expressions in
different forms: write a(x) / b(x) in the form
q(x) + r(x) / b(x), where a(x), b(x), q(x), and
r(x) are polynomials with the degree of r(x)
less than the degree of b(x), using inspection,
long division, or, for more complicated
examples, a computer algebra system.
7. (+) Understand that rational expressions form
a system analogous to the rational numbers,
closed under addition, subtraction,
multiplication, and division by a nonzero
rational expression; add, subtract, multiply,
and divide rational expressions.
Students illustrate how polynomial identities are used to determine
numerical relationships.
Students demonstrate calculations for Pascal’s Triangle to determine
the coefficients of the binomial expansion.
Students use the Binomial Theorem to find the nth term in the
expansion of a binomial to a positive power.
Students rewrite rational expressions, a(x)/b(x), in the form q(x) +
r(x)/b(x) by using factoring, long division, or synthetic division.
Students simplify rational expressions by adding, subtracting,
multiplying, or dividing.
Students understand that rational expressions are closed under
addition, subtraction, multiplication, and division (by a nonzero
expression).
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COMMON CORE CLUSTER
STANDARDS
College and Career Readiness Cluster
Standards for Mathematics
1. Create equations that describe
numbers or relationships.
COMMON CORE STATE STANDARD
Algebra I and II
Linear and quadratic polynomial expressions
are the expectation in Algebra 1 and beyond
quadratic polynomial expressions is the
expectation for Algebra II.
Creating Equations
1. Create equations and inequalities in one
variable and use them to solve problems.
Include equations arising from linear and
quadratic functions, and simple rational and
exponential functions.
2. Create equations in two or more variables to
represent relationships between quantities;
graph equations on coordinate axes with labels
and scales.
3. Represent constraints by equations or
inequalities, and by systems of equations
and/or inequalities, and interpret solutions as
viable or nonviable options in a modeling
context.
4. Rearrange formulas to highlight a quantity of
interest, using the same reasoning as in solving
equations.
WHAT DOES THIS STANDARD MEAN THAT A STUDENT
MUST KNOW AND BE ABLE TO DO?
Students create linear, quadratic, rational and exponential equations
and inequalities in one variable and use them in a contextual
situation to solve problems.
Students create linear and quadratic equations from a set of given
points.
Students create equations in two or more variables to represent
relationships between quantities.
Students graph equations in two variables on a coordinate plane and
label the axes and scales.
Students write and use a system of equations and/or inequalities to
solve real world problems recognizing that equations and
inequalities represent the constraints of the problem.
Students solve multi-variable formulas or literal equations for a
specific variable.
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Mathematics Common Core State Standards
High School Algebra I and II
COMMON CORE CLUSTER
STANDARDS
College and Career Readiness Cluster
Standards for Mathematics
COMMON CORE STATE STANDARD
WHAT DOES THIS STANDARD MEAN THAT A STUDENT
MUST KNOW AND BE ABLE TO DO?
Algebra I students build on experiences graphing and solving
systems of linear equations.
Algebra I and II
Algebra I students focus on and master
A.REI.1 for linear equations and be able to
extend and apply their reasoning to other types
of equations in future courses. Students solve
exponential equations with logarithms in
Algebra II and extend to simple rational and
radical equations.
Reasoning with Equations and Inequalities
1. Understand solving equations as a
1. Explain each step in solving a simple equation Students construct a convincing argument that justifies each step in
process of reasoning and explain the
as following from the equality of numbers
the solution process (assuming an equation has a solution).
reasoning.
asserted at the previous step, starting from the Justifications may include the associative, commutative, and
assumption that the original equation has a
division properties, combining terms, multiplication by the identity.
solution. Construct a viable argument to
justify a solution method.
2. Solve simple rational and radical equations in
one variable, and give examples showing how
extraneous solutions may arise.
2. Solve equations and inequalities in
one variable.
3.Solve linear equations and inequalities in one
variable, including equations with coefficients
represented by letters.
Students solve simple rational and radical equations in one variable,
and identify and explain why they arise.
Students solve linear equations in one variable, including
coefficients represented by letters.
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4a. Use the method of completing the square to
transform any quadratic equations in x into
an equation of the form (x – p)² = q that has
the same solutions. Derive the quadratic
formula from this form.
4b. Solve quadratic equations by inspection (e.g.
for x² = 49), taking square roots, completing
the square, the quadratic formula and
factoring, as appropriate to the initial form of
the equation. Recognize when the quadratic
formula gives complex solutions and write
them as a plus/minus b for real numbers a
and b.
Students transform quadratic equations written in standard form to
an equation in vertex form by completing the square.
Students demonstrate deriving the quadratic formula by completing
the square on the standard form of a quadratic equation.
Students solve quadratic equations in one variable by simple
inspection taking the square root, factoring, and completing the
square.
Students understand why taking the square root of both sides of an
equation yields two solutions.
Students use the quadratic formula to solve any quadratic equation,
recognizing the formula produces all complex solutions.
Students explain how complex solutions affect the graph of a
quadratic equation.
Students solve systems of equations using the elimination method.
3. Solve systems of equations.
5. Prove that, given a system of two equations in
two variables, replacing one equation by the
sum of that equation and a multiple of the other
produces a system with the same solutions.
Students solve a system of equations by substitution.
Students solve a system of equations using matrices.
Students solve systems of equations using graphs.
6. Solve systems of linear equations exactly and
approximately (e.g. with graphs), focusing on
pairs of linear equations in two variables.
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7. Solve a simple system consisting of a linear
equation and a quadratic equation in two
variables algebraically and graphically.
Students solve a system containing a linear equation and a quadratic
equation in two variables graphically and symbolically.
8. (+) Represent a system of linear equations as
a single matrix equation in a vector variable.
Students write a system of linear equations as a single matrix
equation.
9. (+) Students find the inverse of a matrix if it
exists and use it to solve systems of linear
equations.
Students find the inverse of the coefficient matrix in the equation if
it exists.
Students use the inverse of the coefficient matrix to solve the
system.
Students use technology for matrices with dimensions 3 by 3 or
greater.
4. Represent and solve equations and
inequalities graphically.
10. Understand that the graph of an equation in
two variables is the set of all its solutions
plotted in the coordinate plane, often
forming a curve (which could be a line).
Students demonstrate that all solutions to an equation in two
variables are contained on the graph of that equation.
11. Explain why the x-coordinates of the points
where the graphs of the equations y = f(x)
and y = g(x) intersect are the solutions of
the equation f(x) = g(x); find the solutions
approximately, e.g., using technology to
graph the functions, make tables of values,
or find successive approximations. Include
cases where f(x) and/or g(x) are linear,
polynomial, rational, absolute value,
exponential, and logarithmic functions.
Students explain why the interaction of y = f(x) and y = g(x) is the
solution of f(x) = g(x) for any combination of linear, polynomial,
rational, absolute value, exponential, and logarithmic functions.
Students graph the solutions to a linear inequality in two variables as
a half-plane, excluding the boundary for non-inclusive inequalities.
Students graph the solutions set to a system of linear inequalities in
two variables as the intersection of their corresponding half-planes.
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Reformatted Saddlebrook Preparatory School
12. Graph the solutions to a linear inequality in
two variables as a half-plane (excluding the
boundary in the case of a strict inequality)
and graph the solution set to a system of
linear inequalities in two variables as the
intersection of the corresponding halfplanes.
Students graph half-plane two variable linear inequality solutions
and graph a solution set in two variables of the intersection of the
half-planes.
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Reformatted Saddlebrook Preparatory School