Download Essential Skills: Algebra II

Essential Skills:
Algebra II
1) Identify the graphs and equations of the Library of functions.
Learning Goals
A. Students will understand and be able to recognize, interpret, and graph various types of
functions and relations (linear, quadratic, exponential, absolute value, piecewise, step).
B. Students will understand and find a linear equation of best fit and use it to make
predictions.
C. Students will understand and be able to solve and graph inequalities.
2. Do basic operations with functions including composition of functions
Learning Goals
A. Students will understand and use function notation. (Add, subtract, multiply and
divide pairs of functions)
B. Students will understand and use composition of functions and combine functions
by composition.
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3. Solve systems of equations with two or three variables.
Learning Goals
A. Students will understand and use substitution, elimination, and determinants to
solve systems of two or three linear equations in two or three variables.
B. Students will understand and use systems of linear equations and inequalities.
4. Divide and factor polynomials and solve polynomial equations.
Learning Goals
A. Students will understand and be able to divide polynomials by monomials.
B. Students will understand and be able to divide polynomials by other polynomials
using long or synthetic division.
C. Students will understand and be able to solve and factor polynomial equations.
D. Students will understand and be able to use graphing technology to find
approximate solutions for polynomial equations.
E. Students will understand and be able to use polynomial equations to solve word
problems.
F. Students will understand and be able to write a polynomial given its solutions
G. Students will understand and be able to describe the relationships among the
solutions of an equation, the zeros of a function, the x-intercepts of a graph, and
the factors of a polynomial expression.
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5. Do basic operations with rational expressions.
Learning Goals
A. Students will understand and be able to add, subtract, multiply, divide and
simplify algebraic fractions.
B. Students will understand and be able to simplify complex fractions
C. Students will understand and be able to solve equations involving algebraic
fractions
D. Students will understand and be able to solve word problems involving fractional
equations
E. Students will understand and be able to solve problems of direct, inverse, and joint
variation.
6. Simplify radicals and solve radical equations and equations with radicals. Perform basic
operations of exponents including rational and negative exponents.
Learning Goals
A. Students will understand and be able to simplify radicals
B. Students will understand and be able to solve equations that contain radical
expressions.
C. Students will understand and use negative and fractional exponents.
3
7. Add, subtract, multiply, and divide complex numbers.
Learning Goals
A. Student will understand and be able to define complex numbers and perform basic
operations with them.
8. Solve quadratic equations.
Learning Goals
A. Students will understand and be able to factor quadratics.
B. Students will understand and be able to complete the square.
C. Students will understand and be able to use the quadratic formula to find various
types of solutions including complex.
D. Students will understand and graph a quadratic equation.
E. Students will be able to solve word problems using quadratic equations.
F. Students will understand and be able to solve pairs of equations, one quadratic and
one linear or both quadratic.
9. Graph conic sections and identify their equations.
Learning Goals
A. Students will understand and be able to write the equation of conic sections
(circle, ellipse, parabola, and hyperbola.)
B. Students will understand and be able to graph and identify the parts of conic
sections.
4
10.
Solve a logarithmic equation.
Learning Goals
A. Students will understand and be able to use the inverse relationship between
exponents and logarithms.
B. Students will understand and be able to solve logarithmic and exponential
equations and inequalities.
C. Students will understand and be able to use calculators to find decimal
approximations of natural and common logarithmic numeric expressions
11.
Define and use arithmetic and geometric sequences and series.
Learning Goals
A. Students will understand and be able to find terms in sequences.
B. Students will understand and be able to find partial sums of arithmetic and
geometric series.
12.
Use the counting principle to compute combinations, permutations, and probabilities.
Learning Goals
A. Students will understand and be able to compute combinations and permutations.
B. Students will understand and be able to use combinations and permutations to
compute probabilities.
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MICHIGAN CITY HIGH SCHOOL
Algebra II
Ongoing/All Year
1st Quarter
2nd Quarter
3rd Quarter
4th Quarter
Course Title
Assessment
Assessment
Assessment
Assessment
ALGEBRA II
1A, 1B, 2, 3, 4A, QTR1
4B, 4C, 5A, 5B, 6A, QTR2
6B, 7, 8A, 8B, 9A, QTR3
9B, 10, 11, 12, QTR 4
Bundle #
Bundle #
Bundle #
Bundle #
7 Relations & Functions
5 Linear Systems
10 Functions
9 Polynomials
Standards
1.1 1.2 1.3. 1.4. 1.5
1.6 1.7 1.8 2.1 2.2
2.3 5.2 5.3 5.4 5.5
5.6 5.7
Best Practice Methods
9 Polynomials
6 Algebraic Fractions
2 Radicals
Best Practice Methods
Best Practice Methods
Summarizing
Homework & Practice
Frequent Feedback
Advance Organizers
Graphing Technology
Use of Calculators
Reinforcing Effort
Similarities & Differences
Summarizing
Homework & Practice
Frequent Feedback
Advance Organizers
Graphing Technology
Use of Calculators
Reinforcing Effort
Similarities & Differences
Summarizing
Homework & Practice
Frequent Feedback
Advance Organizers
Graphing Technology
Use of Calculators
Reinforcing Effort
Similarities & Differences
3.6
5.6
6.4
Standards
5.2 5.3 5.4
5.7 6.1 6.2
6.5 6.6
3 Complex Numbers
1 Quadratic Equations
4 Conic Sections
5.5
6.3
3.1
4.1
Standards
3.3 3.4 3.5
4.2 5.3
8 Logarithms
11 Sequences & Series
12 Probability
3.7
7.3
8.1
7.4
8.2
Standards
7.5 7.6
8.3 9.1
7.7
9.2
Best Practice Methods
Summarizing
Homework & Practice
Frequent Feedback
Advance Organizers
Graphing Technology
Use of Calculators
Reinforcing Effort
Similarities & Differences
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ALGEBRA II Benchmark #1 – Quadratic Equations
Standard Indicator: Standard 3: Quadratic Equations and Functions; Standard 5:
Polynomials
3.3, 3.4, 3.5, 5.3, 3.7
Declarative Knowledge
Procedural Knowledge
Concepts
Organizing
Ideas
Details
Vocabulary
Various ways to solve quadratic
equations.
Students will understand and be able to:
1. factor quadratics
2. complete square
3. use quadratic formula
4. graph quadratic equation
5. solve word problems
6. systems involving quadratics
Methods of factoring
Complete the square
Find the vertex, axis of symmetry,
minimum/maximum values
Quadratic formula
Processes
Factor out perfect squares
Powers of i.
Using substitution
Completing the square
FOIL
Solving systems of equations
Skills
Simplify radicals
Simplify complex numbers
Creating T-chart
Roots , zeros, solutions, quadratic, vertex,
axis of symmetry, minimum, maximum,
complex numbers, imaginary numbers,
rational & irrational, real numbers,
discriminant, conjugate,
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ALGEBRA II Benchmark #2 - Radicals
Standard Indicator: Standard 3: Quadratic Equations and Functions; Standard 6: Algebraic
Fractions
3.6, 6.1
Declarative Knowledge
Procedural Knowledge
Concepts
Organizing
Ideas
Details
Vocabulary
There are rules used for
Processes
simplifying radicals, solving
radical equations and equations
with radicals.
Students will understand:
1. the laws of exponents
2. simplifying radicals
3. methods for solving
radical equations
4. methods for solving
equations with radicals
Identify parts of rational
Skills
exponent
(numerator/denominator =
power/root)
Rules for negative exponents
Square root, cube root, power,
exponent, index number, radicals,
radicand, fraction, rational
number, irrational number
1. Rules for adding, subtracting, multiplying, and dividing
terms with exponents.
2. Rules for multiplying and dividing radicals.
Prime factorization
Recognition of perfect squares, cubes, powers…
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ALGEBRA II Benchmark #3 – Complex Numbers
Standard Indicator: Standard 3: Quadratic Equations and Functions
3.1
Declarative Knowledge
Procedural Knowledge
Concepts
There are rules for adding,
subtracting, multiplying, and
dividing complex numbers.
Processes
Distributive property
FOIL
Organizing
Ideas
Students will understand:
1. powers of i
2. perform operations with
complex numbers
Details
Powers of i
Using conjugates
Combining like terms
Skills
Apply rules to simplify complex numbers.
Vocabulary
Conjugates, i, imaginary
number(s), complex number(s),
like terms
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ALGEBRA II Benchmark #4 – Conic Sections
Standard Indicator: Standard 4: Conic Sections
4.1, 4.2
Declarative Knowledge
Procedural Knowledge
Concepts
There are unique, but similar,
equations for each conic section.
Processes
Putting equations in standard form.
Use the specific parts to graph the various conic sections
Organizing
Ideas
Students will understand and
recognize the equations for:
parabolas, circles, ellipses, and
hyperbolas.
Various conic sections have specific
parts: asymptotes, center, radius,
axes, foci, vertices…
Skills
Completing the square
Graphing equation
Details
Vocabulary
parabolas, circles, ellipses, and
hyperbolas; asymptotes, center,
radius, axes, foci, vertices, diameter,
directrix, conjugate axis, transverse
axis, conic, major axis, minor axis,
co-vertices,
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ALGEBRA II Benchmark #5 – Linear Systems
Standard Indicator: Standard 2: Linear and Absolute Value Equations and Inequalities;
Standard 3: Quadratic Equations and functions
2.2, 2.3
Declarative Knowledge
Procedural Knowledge
Concepts
There are various ways of solving
systems of equations.
Students will be able to use :
1. substitution
2. elimination
3. multiplication
4. Cramer’s rule
5. graphing
Processes
Setting up a determinant
Details
Students will pick an appropriate
method for solving systems of 2 or
3 variables.
Skills
Vocabulary
Coefficient, variable, matrix,
opposites, determinants, Cartesian
coordinates, elimination,
substitution, Standard Form of
Linear Equation, Slope-Intercept
Form, inequality,
Graphing lines
Common multiples (LCM}
Evaluating a determinant
Substitution & distributive properties of equalities
Other properties of equality/inequalities
Know symbols of inequality(<,>, not equal, …)
Basic operations with matrices
Organizing
Ideas
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ALGEBRA II Benchmark # 6 Algebraic Fractions
Standard Indicator: Standard 6: Algebraic Fractions
6.2, 6.3, 6.4, 6.5, 6.6
Declarative Knowledge
Procedural Knowledge
Concepts
There are rules for simplifying and
solving equation with algebraic
fractions.
Students will be able to
differentiate between direct,
inverse, and joint variation.
Students will be able to simplify
algebraic fractions and solve
algebraic fraction equations.
Processes
Operations with algebraic fractions
Finding the constant of variation and using it to solve
variation problems.
Details
Rules for adding, subtracting,
multiplying and dividing fractions.
Rules for factoring
Skills
Operations with fractions.
Solving proportions—means and extremes.
Factoring.
Vocabulary
Complex fractions, numerator,
denominator, variations(direct,
joint, and inverse), means and
extremes
Organizing
Ideas
12
ALGEBRA II Benchmark # 7 – Relations and Functions
Standard Indicator: Standard 1: Relations and Functions; Standard 2: Linear and Absolute
Value Equations and Inequalities
1.1, 1.4, 1.5, 1.6, 1.7, 1.8, 2.1, 2.4
Declarative Knowledge
Procedural Knowledge
Concepts
There are many kinds of functions
which can be represented by their
own unique equation and graph.
Processes
Organizing
Ideas
Students will identify and graph
various functions.
Details
Functions are to include:
1. linear
2. absolute value
3. step
4. quadratic
5. exponential
6. piece-wise
Vocabulary
Function, step, absolute value,
linear, line, quadratic, exponential,
piece-wise, scatter plots, best fit,
Skills
Recognizing various forms of the functions
Analyzing graphs for their unique form.
Draw and use line of best fit to make a prediction
Write an equation of a line given 2 points.
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ALGEBRA II Benchmark # 8 – Logarithmic Functions
Standard Indicator: Standard 7: Logarithmic & Exponential Functions
7.3, 7.4, 7.5, 7.6, 7.7
Declarative Knowledge
Procedural Knowledge
Concepts
Processes
1.
Rules for converting from one form to another
2.
Rules for switching bases
3.
How to use their calculator to find solutions
4.
How to use a graphing calculator to graph the
functions
5.
Rules for solving equations
Skills
1.
2.
3.
Organizing
Ideas
Details
Vocabulary
There are rules used for solving
logarithmic and exponential
functions.
Students will understand:
1. the inverse relationship
between exponents and
logarithms
2. How to solve exponential
equations
3. how to use a calculator to
find natural & common
logs
Rules for:
1. converting from log form
to exponent form
2. converting from one base
to another
3. using logarithms
Exponent, logarithm, base, power,
inverse, natural log, common log,
e,
Solving equations
Using the Laws of exponents
Graphing equations
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ALGEBRA II Benchmark # 9 - Polynomials
Standard Indicator: Standard 5: Polynomials
5.2, 5.3, 5.4, 5.5, 5.6, 5.7
Declarative Knowledge
Concepts
Organizing
Ideas
Details
Vocabulary
Procedural Knowledge
There are rules for performing operations with
Processes
polynomials. When solving polynomial
equations, there are solutions (roots, zeros) to
the equation.
Students will understand:
1. How to divide polynomials by other
polynomials.
2. How to solve and factor polynomial
equations
3. How to write a polynomial given the
solutions
4. The relationship between: solutions, roots,
zeros, x-intercepts of the graph.
GCF
Skills
FOIL
Synthetic division
Long division
Graphing equations
Zero Product Property
Writing polynomial equations given solutions
Coefficients, descending order, ascending
order, degree of a polynomial or term,
monomial, polynomial, roots, zeros, trinomial,
binomial, solutions, synthetic division
Rules for exponents
Finding the GCF
Dividing by the GCF
Naming coefficients
Finding the degree of a term and a polynomial
Divide using long division
Divide using synthetic division
Using the ZPP to find the zeros
Using the zeros to write the equations
Find the GCF
Distributive Property
FOIL
Graphing
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ALGEBRA II Benchmark #10 – Relations and Functions
Standard Indicator: Standard 1: Relations & Functions
1.2, 1.3
Declarative Knowledge
Procedural Knowledge
Concepts
There are ways of performing
basic operations with functions
including the composition of
functions
Processes
Order of operations within compositions
Organizing
Ideas
Students will understand:
How to add, subtract, multiply,
and divide pairs of functions.
How to find the composition of
functions
Understand function notation: f(x)
=y
Know f _ g = f(g(x))
Know properties of equality
(commutative, associative, etc.) do
not hold for compositions of
functions
Skills
Order of operations
Evaluating expressions
FOIL
Rules of exponents
Combining Like Terms
Details
Vocabulary
Function, commutative property,
associative property, distributive
property, composition,
16
ALGEBRA II Benchmark #11 – Sequences and Series
Standard Indicator: Standard 8: Sequences and Series
8.1, 8.2, 8.3
Declarative Knowledge
Procedural Knowledge
Concepts
There are formulas used to
determine terms of sequences and
sums of series.
Processes
Use the formulas correctly to find the solutions for various
missing terms
Organizing
Ideas
1. Students will understand and be
able to find terms in sequences.
2. Students will understand and be
able to find partial sums of
arithmetic and geometric series
Identify initial (first) term, type of
sequence/series, the common
difference or common ratio.
Recognize summation notation.
Know how to use the formulas.
Skills
How to use technology to assist in finding solutions.
Use the laws (rules) of exponents
Know the order of operations
Solving equations for a given variable (literal equations)
Details
Vocabulary
Sequence, series, geometric,
arithmetic, term, common
difference, common ratio,
summation notation (epsilon),
geometric means, arithmetic
means
17
ALGEBRA II Benchmark #12 – Probability
Standard Indicator: Standard 9: Counting Principles and Probability
9.1, 9.2
Declarative Knowledge
Procedural Knowledge
Concepts
Processes
Use formulas to correctly find the number of possibilities for
an event and the probability that a specific event or series of
events will occur.
Skills
Use factorial function.
Use technology to assist in problem solving.
Use formulas correctly.
Organizing
Ideas
Details
Vocabulary
There are ways to count the number
of possibilities of a given situation,
and determining the probabilities of
a specific event occurring.
Students will understand and be able
to compute combinations and
permutations.
Students will understand and be able
to use combinations and
permutations to compute
probabilities.
1. There are formulas for
determining permutations and
combinations of n distinct objects
taken r at a time.
2. There are formulas for
determining the probability of
events.
Permutation, combination, factorial,
probability, event, odds
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Algebra II Assessment Planning Guide
Essential Outcome 1: Identify the graphs and equations of the Library of functions.
Summative Assessment: Selected Response
Describe assessment and timeline
Method
Knowledge
Types of
Reasoning
Formative:
Students will understand and be able to recognize,
interpret, and graph various types of functions and
relations (linear, quadratic, exponential, absolute
value, piecewise, step).
Students will understand and find a linear equation
of best fit and use it to make predictions.
5+ question
multiple choice
quiz
X
X
5 question
multiple choice
quiz
X
X
Timeline: Two weeks
Formative:
Students will understand and be able to solve and
graph inequalities.
Performance
Skills
Products
Timeline: One week
19
Algebra II Assessment Planning Guide
Essential Outcome 2: Do basic operations with functions including composition of functions
Summative Assessment: Selected Response
Describe assessment and timeline
Formative: Students will
understand and use function
notation. (Add, subtract, multiply
and divide pairs of functions)
Method
10 question
multiple
choice quiz
Knowledge
Types of Reasoning
X
X
Performance Skills
Products
Students will understand and use
composition of functions.
Timeline: ten days
20
Algebra II Assessment Planning Guide
Essential Outcome 3: Solve systems of equations with two or three variables.
Summative Assessment: Selected Response: Use substitution, elimination, and determinants to solve systems of two or three linear
equations in two or three variables. Use systems of linear equations and inequalities. (after 3 weeks)
Describe assessment and timeline
Method
Knowledge
Types of
Performance Products
Reasoning
Skills
Formative: Students will understand and use
Short answer
substitution, elimination, and determinants to solve
systems of two or three linear equations in two or
three variables.
X
X
X
Timeline: Two weeks
21
Algebra II Assessment Planning Guide
Essential Outcome 4: Divide and factor polynomials and solve polynomial equations.
Summative Assessment: divide polynomials by monomials, divide polynomials by other polynomials using long or synthetic
division, solve and factor polynomial equations, use graphing technology to find approximate solutions for polynomial equations,
use polynomial equations to solve word problems, write a polynomial given its solutions, describe the relationships among the
solutions of an equation, the zeros of a function, the x-intercepts of a graph, and the factors of a polynomial expression.
Describe assessment and timeline
Method
Knowledge
Performance Skills
X
Types of
Reasoning
X
Formative: Divide polynomials by
monomials, divide polynomials by
other polynomials using long or
synthetic division.
Timeline: Seven days
Formative: Solve and factor
polynomial equations, use
polynomial equations to solve word
problems. Write a polynomial given
its solutions.
Timeline: Two weeks
Formative: Use graphing
technology to find approximate
solutions for polynomial equations,
Describe the relationships among
the solutions of an equation, the
zeros of a function, the x-intercepts
of a graph, and the factors of a
polynomial expression.
Timeline: One week
Short answer
(5questions)
Short answer
(10 questions)
X
X
X
Short answer, 3
questions (table:
zeros, xintercepts,
factors, etc.)
X
X
X
Products
X
22
Algebra II Assessment Planning Guide
Essential Outcome 5: Do basic operations with rational expressions
Summative Assessment: Students will understand and be able to add, subtract, multiply, divide and simplify algebraic fractions.
Students will understand and be able to simplify complex fractions
Students will understand and be able to solve equations involving algebraic fractions
Students will understand and be able to solve word problems involving fractional equations
Students will understand and be able to solve problems of direct, inverse, and joint variation.
(3 _ weeks total)
Describe assessment and timeline
Method
Knowledge
Types of
Performance
Products
Reasoning
Skills
Formative: Students will understand and be
Multiple
X
X
X
able to add, subtract, multiply, divide and
Choice
simplify algebraic fractions. Students will
understand and be able to simplify complex
fractions.
Timeline: Two weeks
Formative: Students will understand and be
Short Answer X
X
X
able to solve equations involving algebraic
(Work and
fractions, and be able to solve word problems
mixture
involving fractional equations.
problems)
Timeline: One week
23
Algebra II Assessment Planning Guide
Essential Outcome 6: Simplify radicals and solve radical equations and equations with radicals. Perform basic operations of
exponents including rational and negative exponents.
Summative Assessment: Students will understand and be able to simplify radicals
Students will understand and be able to solve equations that contain radical expressions.
Students will understand and use negative and fractional exponents
Describe assessment and timeline
Method
Knowledge Types of Reasoning Performance Skills
Products
Formative: Students will understand
and be able to simplify radicals;
students will understand and use
negative and fractional exponents.
Timeline: Two weeks
Formative: Students will understand
and be able to solve equations that
contain radical expressions.
Timeline: One week
10 question,
short answer
X
X
X
5 questions,
short answer
X
X
X
24
Algebra II Assessment Planning Guide
Essential Outcome 7: Add, subtract, multiply, and divide complex numbers
Summative Assessment: Student will understand and be able to define complex numbers and perform basic operations with them
Describe assessment and timeline
Method
Knowledge
Types of
Performance Products
Reasoning
Skills
Formative: Student will understand and be
Multiple
X
X
X
able to define complex numbers and perform
Choice
basic operations with them.
Timeline: One week
25
Algebra II Assessment Planning Guide
Essential Outcome 8: Solve quadratic equations
Summative Assessment: Factor quadratics, complete the square, use the quadratic formula to find various types of solutions
including complex, graph a quadratic equation, solve word problems using quadratic equations, and solve pairs of equations, one
quadratic and one linear or both quadratic.
Describe assessment and timeline
Method
Knowledge
Types of
Performance Products
Reasoning
Skills
Formative: Students will understand and be able
X
X
X
to factor quadratics, complete the square, and use
the quadratic formula to find various types of
solutions including complex.
Timeline: Two weeks
Formative: Students will understand and graph
X
X
X
a quadratic equation, solve word problems using
quadratic equations, and solve pairs of equations,
one quadratic and one linear or both quadratic.
Timeline: Two weeks
26
Algebra II Assessment Planning Guide
Essential Outcome 9: Graph conic sections and identify their parts
Summative Assessment: Students will understand and be able to write the equation of conic sections (circle, ellipse, parabola, and
hyperbola.). Students will understand and be able to graph and identify the parts of conic sections.
Describe assessment and timeline
Method
Knowledge Types of
Performance Products
Reasoning
Skills
Formative: Students will understand and be able to write Given an
X
X
X
X
the equation of parabolas. Students will understand and be equation,
able to graph and identify the parts of a parabola.
put in the
Timeline: 7 days
correct
form, find
the parts
and graph.
Formative: Students will understand and be able to write Given an
X
X
X
X
the equation of circles, ellipses, & hyperbolas. Students
equation,
will understand and be able to graph and identify the parts put in the
of these conic sections. Decide the conic type.
correct
Timeline: 7 days
form, find
the parts
and graph.
27
Algebra II Assessment Planning Guide
Essential Outcome 10: Solve logarithmic equations
Summative Assessment: Students will understand and be able to use the inverse relationship between exponents and logarithms.
Students will understand and be able to solve logarithmic and exponential equations and inequalities.
Students will understand and be able to use calculators to find decimal approximations of natural and common logarithmic numeric
expressions
Describe assessment and timeline
Method
Knowledge
Types of
Performance Products
Reasoning
Skills
Formative: Students will understand and
Short answer
X
X
X
be able to use the inverse relationship
between exponents and logarithms, solve
logarithmic and exponential equations and
inequalities and be able to use calculators
to find decimal approximations of natural
and common logarithmic numeric
expressions
Timeline: 10 days
28
Algebra II Assessment Planning Guide
Essential Outcome 11: Define and use arithmetic and geometric sequences and series.
Summative Assessment: Students will understand and be able to find terms in sequences. Students will understand and be able to
find partial sums of arithmetic and geometric series.
Describe assessment and timeline
Method
Knowledge Types of
Performance Products
Reasoning
Skills
Formative: Students will understand and be able Short Answer
X
X
X
to find terms in sequences. Students will
understand and be able to find partial sums of
arithmetic and geometric series.
Timeline: 10 days
29
Algebra II Assessment Planning Guide
Essential Outcome 12: Use the counting principle to compute combinations, permutations, and probabilities.
Summative Assessment: Students will understand and be able to compute combinations and permutations. Students will understand
and be able to use combinations and permutations to compute probabilities.
Describe assessment and timeline
Method
Knowledge
Types of
Performance
Products
Reasoning
Skills
Formative: Students will understand and be able Short Answer
X
X
X
to compute combinations and permutations.
Students will understand and be able to use
combinations and permutations to compute
probabilities.
Timeline: 10 days
July 2009
30