Download Algebra I Scope and Sequence

Algebra II
South Central BOCES
Grade HS
UNIT 1 Functional Form and Design (4 weeks)
UNIT 2 Logarithmic Log Jams (4 weeks)
Why are functions necessary to the design and building of skyscrapers?
What is the best way of paying of debt on multiple credit cards?
What financial phenomena can be modeled with exponential and linear functions?
Equations and Inequalities
S.2-GLE.4-EO.a.i
 Create equations and inequalities in one variable and use them to
solve problems.
Solving Systems of Equations
 Solve an equation of the form f(x) = c for a simple function f that
has an inverse and write an expression for the inverse.
 Solve systems of linear equations limited to 3x3 systems exactly
and approximately, focusing on pairs of linear equations in two
variables.
 Solve a simple system consisting of a linear equation and a
quadratic equation in two variables algebraically and graphically.
 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 and
include cases where f(x) and/or g(x) are linear, polynomial,
rational, absolute value, exponential, and logarithmic functions.
S.2-GLE.1EO.e.iii
S.2-GLE.4EO.d.ii
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S.2-GLE.4EO.e.ii
Exponential and Logarithmic Functions
 Use the properties of exponents to transform expressions for
exponential functions with both rational and real exponents.
 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.
 For exponential models, express as a logarithm the solution to
abct = d where a, c, and d are numbers and the base b is 2, 10, or
e; evaluate the logarithm using technology.
 Graph exponential and logarithmic functions, showing intercepts
and end behavior.
 Use the properties of exponents to interpret expressions for
exponential functions.
S.2-GLE.3EO.b.i.3
S.2-GLE.3EO.b.ii
-S.2-GLE.2EO.a.iv
S.2-GLE.1EO.c.iv
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Personal Financial Literacy- Interest Rates, Credit, Lending
 Analyze the impact of interest rates on a personal financial plans.
 Evaluate the costs and benefits of credit.
 Analyze various lending sources, service and financial institutions.
S.2-GLE.2-EO.d.i
S.2-GLE.2EO.d.ii
S.2-GLE.2EO.d.iii
Polynomial, Exponentials, Logarithmic and Trigonometric Functions
 Determine an explicit expression, a recursive process, or steps for
calculation from polynomial, exponentials, logarithmic and
trigonometric contexts.
 Combine polynomial, exponentials, logarithmic and trigonometric
functions using arithmetic operations. (MA10-GR.HS Write arithmetic and geometric sequences both recursively and
with an explicit formula, use them to model situations, and
translate between the two forms. (MA10-GR.HS-)
 Recognize that sequences are functions, sometimes defined
recursively, whose domain is a subset of the integers. (MA10GR.HS-)
 Identify the effect on the graph for polynomial, exponentials,
logarithmic and trigonometric functions of replacing f(x) by f(x) +
k, k f(x), f(kx), and f(x + k) for specific values of k (both positive
and negative); find the value of k given the graphs and
experiment with cases and illustrate an explanation of the effects
on the graph using technology
 For polynomial, exponentials, logarithmic and trigonometric
functions, interpret key features of graphs and tables in terms of
the quantities, and sketch graphs showing key features given a
verbal description of the relationship.
 Calculate and interpret the average rate of change of a function
(presented symbolically or as a table) over a specified interval
and estimate the rate of change from a graph for polynomial,
exponentials, logarithmic and trigonometric functions.
 Compare properties of two functions each represented in a
different way (algebraically, graphically, numerically in tables, or
by verbal descriptions) for polynomial, exponentials, logarithmic
and trigonometric functions.
 Construct linear and exponential functions, including arithmetic
and geometric sequences, given a graph, a description of a
relationship, or two input-output pairs (include reading these
from a table).
 Interpret the parameters in a linear or exponential function in
terms of a context.
S.2-GLE.1EO.d.i.1
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S.2-GLE.1EO.d.ii
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S.2-GLE.1EO.e.i, ii
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S.2-GLE.1EO.c.v.3
S.2-GLE.2EO.a.ii
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Inverse Functions
 Find inverse functions by solving an equation of the form f(x)
= c for a simple function f that has an inverse and write an
expression for the inverse.
S.2-GLE.1EO.e.iii
UNIT 3 Poly Want a Nomial? (4 weeks)
UNIT 4 Radically Rational (4 weeks)
What is the square root of negative 1? What are the implications of having a solution to
this problem?
How did the ancient Greeks multiply binomials and find roots of quadratic equations
without algebraic notations?
Complex Numbers
S.1-GLE.12
EO.c.i
 Know there is a complex number i such that i = –1, and every
S.1-GLE.1complex number has the form a + bi with a and b real.
EO.c.ii
 Use the relation i2 = –1 and the commutative, associative, and
S.1-GLE.1distributive properties to add, subtract, and multiply complex
EO.d.i
numbers
 Solve quadratic equations with real coefficients that have complex
solutions.
How are the models of rational and radical equations related?
Can the graphs of rational and radical functions be transformed in the same way as
quadratic and linear functions?
Rational Expressions
 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 the more
complicated examples, a computer algebra system.
S.2-GLE.3-EO.g
Remainder Theorem
 State and apply the remainder theorem.
S.2-GLE.3EO.d.i)
S.2-GLE.4-EO.b.i
S.2-GLE.4-EO.b.ii
Quadratic, Cubic, and Quartic Polynomials
 Identify zeros of quadratic, cubic, and quartic polynomials when
suitable factorizations are available, and use the zeros to construct
a rough graph of the function defined by the polynomial.
 Prove polynomial identities and use them to describe numerical
relationships.
 Use the structure of a polynomial, rational or exponential
expression to identify ways to rewrite it.
S.2-GLE.3EO.d.ii
S.2-GLE.3EO.e.i
S.2-GLE.3EO.a.ii
Solving Rational and Radical Equations
 Explain each step in solving simple rational or radical equations
as following from the equality of numbers asserted at the
previous step, starting from the assumption that the original
equation has a solution and construct a viable argument to
justify a solution method
 Solve simple rational and radical equations in one variable, and
give examples showing how extraneous solutions may arise.
Rational Exponents
 Explain how the definition of the meaning of rational exponents
follows from extending the properties of integer exponents to
those values, allowing for a notation for radicals in terms of
rational exponents
 Rewrite expressions involving radicals and rational exponents
using the properties of exponents
S.1-GLE.1-EO.a.i
-S.1-GLE.1EO.a.ii
Solve Quadratic Equations
 Solve quadratic equations by inspection, 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 ± bi
for real numbers a and b
Equation of parabola
 Derive the equation of a parabola given a focus and directrix.
S.2-GLE.4EO.c.ii.2, 3
S.4-GLE.3EO.a.3
UNIT 5 Trickster Trigonometry (4 weeks)
UNIT 6 Independently Lucky (3 weeks)
How does the periodicity in the unit circle correspond to the periodicity in graphs of
models of periodic phenomena?
Why can the same class of functions model diverse types of situations (e.g., sales,
manufacturing, temperature, and amusement park rides)?
Unit Circle and Radian Measure
S.2-GLE.1EO.f.i
 Understand radian measure of an angle as the length of the arc on
S.2-GLE.1the unit circle subtended by the angle. (MA10-GR.HS-)
EO.f.ii
 Explain how the unit circle in the coordinate plane enables the
extension of trigonometric functions to all real numbers,
interpreted as radian measures of angles traversed
counterclockwise around the unit circle. (MA10-GR.HS-)
How does probability relate to obtaining car insurance?
Why is it hard for humans to determine if a set of numbers was created randomly?
Trigonometric Functions
 Choose trigonometric functions to model periodic phenomena
with specified amplitude, frequency, and midline.
 Graph trigonometric functions expressed symbolically and show
key features (e.g., period, midline, and amplitude) of the graph, by
hand in simple cases and using technology for more complicated
cases.
S.2-GLE.2EO.c.i
S.2-GLE.2EO.c.iv
Pythagorean Identity
 Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to
find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the
quadrant of the angle. (MA10-GR.HS-S.2-GLE.4-EO.d)
S.2-GLE.4EO.d
Probability
 Describe events as subsets of a sample space (the set of
outcomes) using characteristics (or categories) of the outcomes,
or as unions, intersections, or complements of other events
(“or,” “and,” “not”).
 Understand two events A and B are independent if the
probability of A and B occurring together is the product of their
probabilities, and use this characterization to determine if they
are independent.
 Understand the conditional probability of A given B as P(A and
B)/P(B), and interpret independence of A and B as saying that
the conditional probability of A given B is the same as the
probability of A, and the conditional probability of B given A is
the same as the probability of B.
 Determine if two events are independent by showing that if two
events A and B are independent then the probability of A and B
occurring together is the product of their probabilities.
 Construct and interpret two-way frequency tables of data when
two categories are associated with each object being classified.
Use the two-way table as a sample space to decide if events are
independent and to approximate conditional probabilities.
 Recognize and explain the concepts of conditional probability
and independence in everyday language and everyday
situations.
 Find the conditional probability of A given B as the fraction of B’s
outcomes that also belong to A, and interpret the answer in
terms of the model.
 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and
interpret the answer in terms of the model.
S.3-GLE.3EO.a.i
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S.3-GLE.3EO.a.ii
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S.3-GLE.3EO.b.i
S.3-GLE.3EO.b.ii
Personal Financial Literacy - Insurance and Risk
 Analyze the cost of insurance as a method to offset the risk of a
situation.
S.3-GLE.3-EO.c
UNIT 7 Survey Says… (3 weeks)
When should sampling be used? When is sampling better than a census?
Sampling, Data, and Surveys
 Understand statistics as a process for making inferences about
population parameters based on a random sample from that
population.
 Decide if a specified model is consistent with results from a given
data-generating process, e.g., using simulation.
 Recognize the purposes of and differences among sample surveys,
experiments, and observational studies and explain how
randomization relates to each.
 Use data from a sample survey to estimate a population mean or
proportion; develop a margin of error through the use of
simulation models for random sampling.
 Use data from a randomized experiment to compare two
treatments; use simulations to decide if differences between
parameters are significant.
 Evaluate reports based on data.
S.3-GLE.2EO.a.i)
S.3-GLE.2EO.a.ii
S.3-GLE.2EO.b.i
S.3-GLE.2EO.b.ii, iii
S.3-GLE.2EO.b.iv
S.3-GLE.2EO.b.vi
Mean and Standard Deviation
 Use the mean and standard deviation of a data set to fit it to a
normal distribution and to estimate population percentages and
recognize that there are data sets for which such a procedure is
not appropriate; use calculators, spreadsheets, and tables to
estimate areas under the normal curve.
S.3-GLE.1EO.a.iv, v