Download Algebra II

Algebra 2
Conceptual Category(s)
Domain
Cluster
Alignments:
CCSS: See below
Performance: 1.6, 3.4
Knowledge: (MA) 4
MACLE: See below
NETS: 1d; 4c,d
DOK: 1-3
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Algebra
Functions
Seeing Structure in Expressions (A-SSE)
Reasoning with Equations and Inequalities (A-REI)
Arithmetic with Polynomials and Rational Expressions (A-APR)
Creating Equations (A-CED)
Building Functions (F-BF)
Interpret the structure of expressions (A-SSE.1b, 2)
Understand solving equations as a process of reasoning and explain the reasoning (A-REI.2)
Represent and solve equations and inequalities graphically (A-REI.11)
Rewrite rational expressions (A-APR. 7)
Create equations that describe numbers or relationships (A-CED.1-3)
Build new functions from existing functions (F-BF.4a,b)
Standards
A-SSE.1b, 2
1. Interpret expressions that represent a quantity in terms of its
context ★
b. Interpret complicated expressions by viewing one or more of
their parts as a single entity. For example, interpret P(1+r)n as
the product of P and a factor not depending on P
2. Use the structure of an expression to identify ways to rewrite it.
For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a
difference of squares that can be factored as (x2 – y2)(x2 + y2)
Learning Targets
Unit A: Rational Expressions, Equations, & Inequalities: Perform
operations with and simplify rational expressions, create and solve rational
equations and inequalities, and find inverses of rational functions
Write equivalent expressions in a variety of forms by factoring
Factor GCF, trinomials, difference of squares, sums & differences of
cubes, and factor by grouping
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CCSS: A-SSE.1b,2
MACLE: AR.1.C; AR.2.B
A-REI.2, 11
2. Solve simple rational and radical equations in one variable, and
give examples showing how extraneous solutions may arise
Board Approved 7-15-13
Revised 2013
1
Algebra 2
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
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A-APR.7
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
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A-CED.1-3
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. For example,
represent inequalities describing nutritional and cost constraints
on combinations of different foods
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F-BF. 4a,b
4. Find inverse functions
a. 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. For
example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1
b. (+) Verify by composition that one function is the inverse of
another
Add and subtract rational expressions
Multiply and divide rational expressions
Simplify rational expressions including complex rational fractions
Rewrite rational expressions in different forms
CCSS: A-APR.7
MACLE: N/A
Create a simple rational equation/inequality in 1 variable
Solve a simple rational equation/inequality
CCSS: A-CED.1; A-REI.2
MACLE: AR.1.D; AR.2.C
Solve a system of functions using graphing calculator technology
including linear, quadratic, and rational functions
Verify the solution(s) of a system of functions
Explain the solution(s) of a system of functions
Isolate any variable in a rational equation
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CCSS: A-CED.1; A-REI.2,11
MACLE: AR.1.D; AR.2.C
Determine an equation in two variables using provided data (word
problems)
Recognize viable solutions
Understand/apply restrictions on domain
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CCSS: A-CED.2,3
MACLE: N/A
Find the inverse of a rational function
Verify inverses through function composition
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CCSS: F-BF.4a,b
MACLE: AR.2.B
Board Approved 7-15-13
Revised 2013
2
Algebra 2
Instructional Strategies
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Lecture enhanced with:
• SMART Notebook
• PowerPoint
• the Internet
Drill and guided practice
Problem solving: Domain investigation
Reflective discussion
Student self-reflection
Class discussion
Computer assisted instruction
Game: Factoring Match-It
Assessments/Evaluations
The students will be assessed on the concepts taught using a variety of modalities:
• Direct teacher observations
• Project: Potential Project: Light it Up
• Quizzes
• Homework assignments – see pacing guide
• Formal common assessment – Unit A test
Mastery Level: 80%
Board Approved 7-15-13
Revised 2013
3
Algebra 2
Instructional Resources/Tools
Textbook(s): (sample copies on the bookshelf)
• Pearson – Algebra 2 Common Core (primary source)
• Holt – Algebra 2 Common Core Edition
• Houghton Mifflin – On Core Mathematics: Algebra 2
• Website(s):
• http://www.mathsisfun.com/algebra/rational-expression.html
• http://www.purplemath.com/modules/rtnldefs.htm
• http://www.youtube.com/watch?v=B4bVlDgHF5I
• Graphing calculator
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Board Approved 7-15-13
Revised 2013
4
Algebra 2
Conceptual
Category(s)
Domain
Cluster
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Algebra
Number and Quantity
Functions
Arithmetic with Polynomials and Rational Expressions (A-APR)
Seeing Structures of Expressions (A-SSE)
Creating Equations (A-CED)
Reasoning with Equations and Inequalities (A-REI)
Building Functions (F-BF)
Interpreting Functions (F-IF)
Linear, Quadratic, and Exponential Models (F-LE)
Complex Number System (N-CN)
Perform arithmetic operations on polynomials (A-APR.1)
Understand the relationship between zeros and factors of polynomials (A-APR.2, 3)
Use polynomial identities to solve problems (A-APR.5)
Rewrite rational expressions (A-APR.6)
Interpret the structures of expressions (A-SSE.1, 2)
Create equations that describe numbers or relationships (A-CED.2, 3)
Represent and solve equations and inequalities graphically (A-REI.11)
Build new functions from existing functions (F-BF.3)
Analyze functions using different representations (F-IF.7)
Construct and compare linear, quadratic, and exponential models and solve problems (F-LE.3)
Use complex numbers in polynomial identities and equations (N-CN.8)
Alignments:
CCSS: See below
Performance: 3.4
Knowledge: (MA) 4
MACLE: See below
NETS: 3d; 5b; 6b
DOK: 1-3
Board Approved 7-15-13
Revised 2013
5
Algebra 2
Standards
A-APR.1-3, 5
1. Understand that polynomials form a system analogous to the
integers, namely, they are closed under the operations of addition,
subtraction, and multiplication; add, subtract, and multiply
polynomials
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 zeros of polynomials when suitable factorizations are
available, and use the zeros to construct a rough graph of the
function defined by the polynomial
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
6. Rewrite simple rational expressions in different forms; write
a(x)/b(x) in the form q(x) + r(x)/b(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
A-SSE.1, 2
1. Interpret expressions that represent a quantity in terms of its
context★
2. Use the structure of an expression to identify ways to rewrite it.
For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a
difference of squares that can be factored as (x2 – y2)(x2 + y2)
Learning Targets
Unit B: Polynomial Expressions & Functions: Simplify polynomial
expressions, solve problems with polynomials, and graph and identify
characteristics of polynomial functions
Add, subtract, and multiply polynomials
Classify and graph polynomials
Use properties of end behavior to analyze, describe, and graph
polynomial functions
Identify maxima and minima of polynomial functions (turning points)
Determine the intervals when a polynomial function is increasing or
decreasing
Identify even and odd functions from a graph and/or algebraic
expression
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CCSS: A-APR.1; F-IF.7; F-LE.3
MACLE: AR.2.A
Use the Factor Theorem to determine factors and zeros of a polynomial
Identify the multiplicity of zeros
Write a polynomial function from its zeros
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CCSS: A-APR.3; A-SSE.2
MACLE: AR.2.A
Find all the zeros of a polynomial by factoring (GCF, quadratic
trinomials, difference of squares, grouping, sum or difference of cubes)
Find all the real zeros of a polynomial by graphing
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A-CED.2, 3
2. Create equations in two or more variables to represent
relationships between quantities; graph equations on coordinate
axes with labels and scales
CCSS: A-SSE.2; A-APR.3
MACLE: AR.2.A
Use long division and synthetic division to divide polynomials.
Revised 2013
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Board Approved 7-15-13
6
Algebra 2
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. For example,
represent inequalities describing nutritional and cost constraints
on combinations of different foods
Know and apply the Remainder Theorem
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CCSS: A-APR.2
MACLE: NO.1.B
Use the Rational Root Theorem and the Conjugate Root Theorem to
solve polynomial equations
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A-REI.11
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 ★
CCSS: N-NC.8
MACLE: AR.2.C
Use the Fundamental Theorem of Algebra to solve polynomial
equations with complex solutions
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CCSS: N.NC.8; A-APR.2,3
MACLE: NO.1.B
F-BF.3
3. Identify the effect on the graph of replacing f(x) by f(x)+k, kf(x),
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f(kx), and f(x+k) for specific values of k (both positive and
negative); find the value of k given the graphs. Experiment with
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cases and illustrate an explanation of the effects on the graph using
technology. Include recognizing even an odd functions from their
graphs and algebraic expressions for them
F-IF. 7
7. Graph functions expressed symbolically and show key features of
the graph, by hand in simple cases and using technology for more
complicated cases ★
F-LE. 3
3. Observe using graphs and tables that a quantity increasing
exponentially eventually exceeds a quantity increasing linearly,
quadratically, or (more generally) as a polynomial function
N-CN. 8
8. (+) Extend polynomial identities to the complex numbers. For
example, rewrite x2 + 4 as (x + 2i)(x – 2i)
Use binomial expansion to expand binomial expressions that are raised
to positive integer powers
Determine the coefficients of a binomial expansion by Pascal’s Triangle
CCSS: A-APR.1,5
MACLE: AR.1.C
Use technology to find polynomial models for a given set of data
(linear, quadratic, cubic, quartic)
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CCSS: A-CED.3; F-IF.7; A-CED.2; A-SSE.1
MACLE: DP.2.C
Apply transformations to graphs of polynomials
Use technology to find the intersection of graphs (linear vs. polynomial)
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CCSS: F-BF.3; F-IF.7; A-REI.11
MACLE: GSR.3.B
Board Approved 7-15-13
Revised 2013
7
Algebra 2
Instructional Strategies
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Lecture enhanced with:
• SMART Notebook
• PowerPoint
• the Internet
Drill and guided practice
Demonstrations: Investigate the graphs and characteristics of f(x) = xn)
Problem solving: Build a box of maximum volume
Reflective discussion
Student self-reflection
Class discussion
Technology enhanced (TI graphing calculator)
Assessments/Evaluations
The students will be assessed on the concepts taught using a variety of modalities:
• Direct teacher observations
• Quizzes
• Homework assignments – see pacing guide
• Formal common assessment – Unit B test
Mastery Level: 80%
Instructional Resources/Tools
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Textbook(s): (sample copies on bookshelf)
• Pearson – Algebra 2 Common Core (primary source)
• Holt – Algebra 2 Common Core Edition
• Houghton Mifflin – On Core Mathematics: Algebra 2
Graphing calculator
Board Approved 7-15-13
Revised 2013
8
Algebra 2
Conceptual Category(s)
Domain
Cluster
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Algebra
Functions
The Real Number System (N-RN)
Building Functions (F-BF)
Reasoning with Equations and Inequalities (A-REI)
Creating Equations (A-CED)
Interpreting Functions (F-IF)
Extend the properties of exponents to rational exponents (N-RN.2)
Build a function that models a relationship between two quantities (F-BF.1)
Build new functions from existing functions (F-BF.3, 4)
Understand solving equations as a process of reasoning and explain the reasoning (A-REI.2)
Represent and solve equations and inequalities graphically (A-REI.11)
Create equations that describe numbers or relationships (A-CED.2, 3)
Analyze functions using different representations (F-IF.7)
Alignments:
CCSS: See below
Performance: 1.4, 1.6, 2.7, 3.4, 3.6, 3.7
Knowledge: (MA) 1,4,5
MACLE: See below
NETS: 1a; 3d; 5c
DOK: 1-3
Standards
N-RN.2:
2. Rewrite expressions involving radicals and rational exponents
using the properties of exponents
F-BF.1c, 3, 4a,b,d
1. Write a function that describes a relationship between two
quantities ★
c. (+) Compose functions. For example, if T(y) is the temperature
in the atmosphere as a function of height, and h(t) is the height
of a weather balloon as a function of time, then T(h(t)) is the
temperature at the location of the weather balloon as a
function of time
Learning Targets
Unit C: Radical Expressions & Functions: Perform operations with and
simplify radical expressions; create and solve radical equations and
inequalities; graph and transform radical functions; find inverses and the
composition of radical functions
Change radical expressions to expressions with fractional exponents
Change expressions with fractional exponents to radical expressions
Evaluate expressions with rational exponents or nth roots
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CCSS: N-RN.2
MACLE: AR.2.A; NO.3.D
Board Approved 7-15-13
Revised 2013
9
Algebra 2
3. Identify the effect on the graph 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. Experiment with
cases and illustrate an explanation of the effects on the graph using
technology. Include recognizing even and odd functions from their
graphs and algebraic expressions for them
4. Find inverse functions
a. 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. For
example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1
b. (+) Verify by composition that one function is the inverse of
another
d. (+) Produce an invertible function from a non-invertible
function by restricting the domain
A-REI.2, 11
2. Solve simple rational and radical equations in one variable, and
give examples showing how extraneous solutions may arise
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
A-CED.2, 3
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. For example,
represent inequalities describing nutritional and cost constraints
on combinations of different foods
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Solve equations involving nth roots
Solve radical equations that may include extraneous solutions
CCSS: A-REI.2
MACLE: AR.2.A; NO.3.D
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Perform operations with functions (add, subtract, multiply, & divide)
Perform composition on given functions (ex: f(g(x)) )
CCSS: F-BF.1c
MACLE: AR.1.B; AR.2.B
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Perform composition on given functions (ex: f(g(x)) )
Determine the domain of a composition
CCSS: F-BF.1c; A-CED.3
MACLE: AR.2.B
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Find the inverse of a function, including power functions
Determine whether two functions are inverses
Determine if a function is one to one
Write an inverse function of a non-invertible function by restricting the
domain
CCSS: F-BF.4a,b,d
MACLE: AR.2.B
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Graph square root functions by hand and using a calculator
Identify and describe transformations of f(x) including
horizontal/vertical reflection, horizontal/vertical stretch and shrink,
horizontal/vertical shift
Determine the domain and range of square root functions
Identify even and odd functions from graph and algebraic expressions
CCSS: F-BF.3; F-IF.7; A-CED.3
MACLE: AR.2.A
Board Approved 7-15-13
Revised 2013
10
Algebra 2
F-IF.7b
7. Graph functions expressed symbolically and show key features of
the graph, by hand in simple cases and using technology for more
complicated cases ★
b. Graph square root, cube root, and piecewise-defined functions,
including step functions and absolute value functions
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Graph cube root functions by hand and using a calculator
Identify and describe transformations of f(x) including
horizontal/vertical reflection, horizontal/vertical stretch and shrink,
horizontal/vertical shift
Determine the domain and range of cube root functions
Identify even and odd functions from graph and algebraic expressions
CCSS: F-BF.3; F-IF.7; A-CED.3
MACLE: AR.2.A
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Solve a system of functions (by hand and using a calculator)
Verify the solution(s)
Explain the meaning of the solution(s)
Determine a radical equation in two variables using provided data (word
problems)
Graph equations on a coordinate plane including labels and scales on
axes
Identify an appropriate domain and range for a given model involving
radical functions including real world application problems
CCSS: A-REI.11; A-CED.2,3
MACLE: AR.1.C,E; AR.2.C; AR.3.A
Instructional Strategies
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Lecture enhanced with:
• SMART Notebook
• PowerPoint
• the Internet
Drill and guided practice
Demonstrations: Use the graphing calculator to investigate square root and cube root transformations
Reflective discussion
Student self-reflection
Class discussion
Computer assisted instruction
Board Approved 7-15-13
Revised 2013
11
Algebra 2
Assessments/Evaluations
The students will be assessed on the concepts taught using a variety of modalities:
• Direct teacher observations
• Quizzes
• Homework assignments – see pacing guide
• Formal common assessment – Unit C test
Mastery Level: 80%
Instructional Resources/Tools
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Textbook(s): (sample copies on bookshelf)
• Pearson – Algebra 2 Common Core (primary source)
• Holt – Algebra 2 Common Core Edition
• Houghton Mifflin – On Core Mathematics: Algebra 2
Website(s): http://www.purplemath.com/modules/graphrad.htm
Graphing calculator
Board Approved 7-15-13
Revised 2013
12
Algebra 2
Conceptual Category(s)
Domain
Cluster
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Number & Quantity
Algebra
Functions
The Real Number System (N-RN)
Seeing Structure in Expressions (A-SSE)
Creating Equations (A-CED)
Reasoning with Equations and Inequalities (A-REI)
Interpreting Functions (F-IF)
Building Functions (F-BF)
Linear, Quadratic, and Exponential Models (F-LE)
Extend the properties of exponents to rational exponents (N-RN.1, 2)
Interpret the structure of expressions (A-SSE.1a,b, 2)
Write expressions in equivalent forms to solve problems (A-SSE.3c)
Create equations that describe numbers or relationships (A-CED.1-3)
Represent and solve equations and inequalities graphically (A-REI.10, 11)
Understand the concept of a function and use function notation (F-IF.1, 2)
Interpret functions that arise in applications in terms of the context (F-IF.4-6)
Analyze functions using different representations (F-IF.7e, 8b)
Build a function that models a relationship between two quantities (F-BF.1a,b)
Build new functions from existing functions (F-BF.3)
Construct and compare linear, quadratic, and exponential models and solve problems (F-LE.1a,c, 2)
Interpret expressions for functions in terms of the situation they model (F-LE.5)
Alignments:
CCSS: See below
Performance: 1.6, 1.8, 1.10
Knowledge: (MA) 4
MACLE: See below
NETS: 1d; 4a; 6a
DOK: 1-3
Board Approved 7-15-13
Revised 2013
13
Algebra 2
Standards
N-RN.1, 2:
1. Explain how the definition of the meaning of rational exponents
follow from extending the properties of integer exponents to those
values, allowing for a notation for radicals in terms of rational
exponents. For example, we define 51/3 to be the cube root of 5
because we want (51/3)3 to hold, so (51/3)3 must equal 5
2. Rewrite expressions involving radicals and rational exponents
using the properties of exponents
A-SSE.1a,b, 3c:
1. Interpret expressions that represent a quantity in terms of its
context
a. Interpret parts of an expression, such as terms, factors, and
coefficients ★
b. Interpret complicated expressions by viewing one or more of
their parts as a single entity. For example, interpret P(1+r)n
as the product of P and a factor not depending on P ★
3. Choose and produce an equivalent form of an expression to reveal
and explain properties of the quantity represented by the
expression
c. Use properties of exponents to transform expressions for
exponential functions. For example, the expression 1.15t can
be rewritten as (1.151/12)12t to reveal he approximate
equivalent monthly interest rate if the annual rate is 15%★
A-CED.1-3:
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 ★
Learning Targets
Unit D: Exponential Functions: Write, solve, and graph exponential
functions
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Review properties of exponents to rewrite expressions involving
radicals and rational exponents
Identify parts of an exponential expression/equation
Identify exponential equations as growth, decay, or neither
Prove that exponential functions grow by equal factors over equal
intervals
CCSS: N-RN.1,2; F-IF.6,8b; A-SSE.1a,b; F-LE.1a
MACLE: AR.2.A; AR.4.A
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Graph exponential growth functions
Identify asymptotes, domain/range, intercepts, end behavior,
increasing/decreasing & transformations
Understand that the graph of an equation in two variables is the set of all
its solutions plotted in the coordinate plane
CCSS: A-REI.10; A-CED.2,3: F-IF.4,7e; F-BF.3
MACLE: AR.4.A
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Graph exponential decay functions
Identify asymptotes, domain/range, intercepts, end behavior,
increasing/decreasing & transformations
Understand that the graph of an equation in two variables is the set of all
its solutions plotted in the coordinate plane
Recognize situations in which a quantity grows or decays by a constant
percent rate per unit interval relative to another
CCSS: A-REI.10;A-CED.2,3: F-IF.4, F-IF.5,7e; F-BF.3; F-LE.1c
MACLE: AR.4.A
Board Approved 7-15-13
Revised 2013
14
Algebra 2
3. Represent constraints by equations or inequalities, and by systems
of equations and/or inequalities, and interpret solutions as viable
or nonviable options in modeling context. For example, represent
inequalities describing nutritional and cost constraints on
combinations of different foods ★
A-REI.10:
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)
F-IF.1, 2, 4-7e, 8b:
1. Understand that a function from one set (called the domain) to
another set (called the range) assigns to each element of the
domain exactly one element of the range. If f is a function and x is
an element of its domain, then f(x) denotes the output of f
corresponding to the input x. The graph of f is the graph of the
equation y = f(x)
2. Use function notation, evaluate functions for inputs in their
domains, and interpret statements that use function notation in
terms of a context
4. For a function that models a relationship between two quantities,
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. Key features include: intercepts;
intervals where the function is increasing, decreasing, positive, or
negative; relative maximums and minimums; symmetries; end
behavior; and periodicit★
5. Relate the domain of a function to its graph and, where applicable,
to the quantitative relationship it describes. For example, if the
function h(n) gives the number of person-hours it takes to
assemble n engines in a factory, then the positive integers would
be an appropriate domain for the functions ★
6. Calculate and interpret the average rate of change of a function
(presented symbolically or as a table) over a specified interval.
Estimate the rate of change from a graph ★
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Write and solve exponential growth and decay problems from a set of
points
Construct exponential functions given a graph or two input-output pairs
CCSS: A-SSE.3c; A-CED.1-3; F-LE.2; F-IF.1,2
MACLE: AR.3.A
•
•
•
Use and solve exponential growth and decay equations and inequalities
to model real-life situations
Construct exponential functions given a description of a relationship
Interpret the parameters of an exponential function in terms of context
CCSS: F-BF.1b; F.LE.2,5; A-CED.1,3
MACLE: AR.4.A
Board Approved 7-15-13
Revised 2013
15
Algebra 2
7. Graph functions expressed symbolically and show key features of
the graph, by hand in simple cases and using technology for more
complicated cases
e. Graph exponential and logarithmic functions, showing
intercepts and end behavior, and trigonometric functions,
showing period, midline, and amplitude
8. Write a function defined by an expression in different but
equivalent forms to reveal and explain different properties of the
function
b. Use properties of exponents to interpret expressions for
exponential functions. For example, identify percent of rate of
change in functions such as y = (1.02)t, y = (0.97)t, y =
(1.01)12t, y = (1.2)t/10, and classify them as representing
exponential growth or decay
F-BF.1b, 3:
1. Write a function that describes a relationship between two
quantities
b. Combine standard function types using arithmetic operations.
For example, build a function that models the temperature of a
cooling body by adding a constant function to a decaying
exponential, and relate these functions to the model
3. Identify the effect of the graph of replacing f(x) by f(x) + k, k f(x),
and f(x + k) for specific values of k (both positive and negative);
find the value of k given the graphs. Experiment with cases and
illustrate and explanation of the effects on the graph using
technology. Include recognizing even and odd functions from
their graphs and algebraic expressions for them
Board Approved 7-15-13
Revised 2013
16
Algebra 2
F-LE.1a,c, 2, 5:
1. Distinguish between situations that can be modeled with linear
functions and with exponential functions
a. Prove that linear functions grow by equal differences over
equal intervals, and that exponential functions grow by equal
factors over equal intervals ★
c. Recognize situations in which a quantity grows or decays by
constant percent rate per unit interval relative to another ★
2. Construct linear and exponential functions, including arithmetic
and geometric sequences, given a graph, a description of a
relationship, or two input-output pairs (including reading these
from a table) ★
5. Interpret the parameters in a linear or exponential function in
terms of a context
Instructional Strategies
Direct and explicit instruction
Guided and independent practice
Infuse AFL tenets in instruction
Graphic organizer – Exponential Growth/Decay
Small-group instruction/facilitation
Summarizing/synthesizing concepts
Student self-reflection
Teacher self-reflection
Class discussion
• Game: Exponent Block Review
•
•
•
•
•
•
•
•
•
Board Approved 7-15-13
Revised 2013
17
Algebra 2
Assessments/Evaluations
The students will be assessed on the concepts taught using a variety of modalities:
• Direct teacher observations
• Quizzes
• Homework assignments – see pacing guide
• Formal common assessment – Unit D test
Mastery Level: 80%
Instructional Resources/Tools
•
•
•
•
•
Textbook(s): (sample copies on bookshelf)
• Pearson – Algebra 2 Common Core (primary source)
• Holt – Algebra 2 Common Core Edition
• Houghton Mifflin – On Core Mathematics: Algebra 2
Website(s) such as khanacademy.org
Graphing calculator
Computer simulation software
Exponential functions apps
Board Approved 7-15-13
Revised 2013
18
Algebra 2
Conceptual Category(s)
Domain
Cluster
Alignments:
CCSS: See below
Performance: 3.4, 1.6
Knowledge: (MA) 4
MACLE: See below
NETS: 3d; 6a,b
DOK: 1-3
•
•
•
•
•
•
•
•
•
•
•
•
Algebra
Functions
Creating Equations (A-CED)
Reasoning with Equations and Inequalities (A-REI)
Interpreting Functions (F-IF)
Building Functions (F-BF)
Linear, Quadratic, and Exponential Models (F-LE)
Create Equations that describe numbers or relationships (A-CED.2-4)
Represent and solve equation and inequalities graphically (A-REI.11)
Analyze functions using different representations (F-IF.7e)
Build new functions from existing functions (F-BF.4a,b, 5)
Construct and compare linear, quadratic, and exponential models and solve problems (F-LE.4)
Standards
Learning Targets
A-CED.2-4
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. For example,
represent inequalities describing nutritional and cost constraints
on combinations of different foods
4. Rearrange formulas to highlight a quantity of interest, using the
same reasoning as in solving equations. For example, rearrange
Ohm’s law V=IR to highlight resistance R
Unit E: Logarithmic Functions: Create and solve logarithmic equations
and inequalities; graph and transform log functions by hand and using a
calculator; understand the inverse relationship between exponents and logs;
and solve a system of equations using a variety of functions
•
•
•
Find the inverse of a logarithmic equation and identify it as an
exponential equation
Convert equations to/from exponential to/from logarithmic form
Verify by composition that exponential and logarithmic functions are
inverses
CCSS: F-BF.4a,b, 5
MACLE: AR.1.D; AR.2.A,B
Board Approved 7-15-13
Revised 2013
19
Algebra 2
A-REI.11
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 of the approximately; e.g.,
using technology to graph the functions, or find successive
approximations. Include cases where f(x) and/or g(x) are linear,
polynomial, rational, absolute value, exponential, and logarithmic
functions
F-IF.7e
7. Graph functions expressed symbolically and show key features of
the graph, by hand in simple cases and using technology for more
complicated cases
e. Graph exponential and logarithmic functions, showing
intercepts and end behavior, and trigonometric functions,
showing period, midline, and amplitude
F-BF.4a,b, 5
4. Find inverse functions
a. 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. For
example, f(x) =2x3 of f(x) = (x+1)/(x-1) for x≠1
b. (+) Verify by composition that one function is the inverse of
another
5. (+) Understand the inverse relationship between exponents and
logarithms and use this relationship to solve problem involving
logarithms and exponents
F-LE.4
4. 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
•
•
Solve logarithmic equations algebraically using properties of logarithms
Solve logarithmic equations without a calculator
CCSS: F-BF.4a, 5; F-LE.4
MACLE: AR.1.D; AR.2.A,B
•
Evaluate logarithms and solve logarithmic equations with base 2, 10,
and e
CCSS: F-LE.4
MACLE: AR.1.D; AR.2.A
•
Graph logarithmic functions by hand including identifying the domain,
range, transformations, and asymptote including labels and scales on
axes
CCSS: A-CED.2; F-IF.7e
MACLE: AR.1.C; AR.2.A; AR.3.A; AR.4.A; GSR.4.B
•
•
Determine a logarithmic equation with two variables using provided
data such as word problems
Identify an appropriate domain and range for a given model involving
logarithmic functions
CCSS: A-CED.2,3
MACLE: AR.2.A,D; AR.3.A; GSR.4.B
•
Isolate one variable in the logarithmic equation and solve the
logarithmic equation using a calculator
CCSS: A-CED.4; F-LE.4
MACLE: AR.1.D; AR.2.A,B
•
•
Solve a system of functions (by calculator and by hand)
Verify the solution(s), and explain the meaning of the solution(s)
CCSS: A-REI.11
MACLE: AR.2.D; AR.4.A; GSR.4.B; M.2.D
Board Approved 7-15-13
Revised 2013
20
Algebra 2
Instructional Strategies
•
•
•
•
•
•
•
•
Lecture enhanced with SMART Notebook
Drill and guided practice
Problem solving using application problems such as:
• exponential growth/decay
• compound/continuous interest
Reflective discussion
Student self-reflection
Class discussion
Computer assisted instruction
Game: MATHO review game
Assessments/Evaluations
The students will be assessed on the concepts taught using a variety of modalities:
• Direct teacher observations
• Quizzes
• Homework assignments – see pacing guide
• Formal common assessment – Unit E test
Mastery Level: 80%
Instructional Resources/Tools
•
•
•
Textbook(s): (sample copies on bookshelf)
• Pearson – Algebra 2 Common Core (primary source)
• Holt – Algebra 2 Common Core Edition
• Houghton Mifflin – On Core Mathematics: Algebra 2
Website(s): http://www.sparknotes.com/math/algebra2/logarithmicfunctions/
Graphing calculator
Board Approved 7-15-13
Revised 2013
21
Algebra 2
Conceptual Category(s)
Domain
Cluster
•
•
•
•
•
•
•
•
•
•
Algebra
Functions
Creating Equations (A-CED)
Reasoning with Equations and Inequalities (A-REI)
Interpreting Functions (F-IF)
Building Functions (F-BF)
Represent and solve equations and inequalities graphically (A-REI.11)
Analyze functions using different representations (F-IF.7b, 9)
Build new functions from existing functions (F-BF.3)
Create equations that describe numbers or relationships (A-CED.2, 3)
Alignments:
CCSS: See below
Performance: 3.4, 3.5
Knowledge: (MA) 4
MACLE: See below
NETS: 1b; 3a; 4d
DOK: 1-3
Standards
Learning Targets
A-REI.11
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
Unit F: Other Functions: Graph, compare, write and solve piecewise, step
and absolute value functions
F-IF.7b, 9
7. Graph functions expressed symbolically and show key features of
the graph, by hand in simple cases and using technology for more
complicated cases ★
b. Graph square root, cube root, and piecewise-defined functions,
including step functions and absolute value functions
•
•
•
•
Graph piecewise functions (by hand and on the graphing calculator) on
a coordinate plane including labels and scales on axes
Graph absolute value functions (by hand and on the graphing calculator)
on coordinate plane including labels and scales on axes
Identify and describe transformations of the functions including
horizontal/vertical reflection, horizontal/vertical stretch and shrink,
horizontal/vertical shift
State the domain/range
CCSS: F-IF.7b; F-BF.3; A-CED.2,3
MACLE: AR.1.E; AR.2.C
Board Approved 7-15-13
Revised 2013
22
Algebra 2
9. Compare properties of two functions each represented in a
different way (algebraically, graphically, numerically in tables, or
by verbal descriptions). For example, given a graph of one
quadratic function and an algebraic expression for another, say
which has the larger maximum
•
•
F-BF.3
3. Identify the effect on the graph 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. Experiment with
cases and illustrate an explanation of the effects on the graph using
technology. Include recognizing even and odd functions from their
graphs and algebraic expressions for them
•
A-CED.2, 3
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. For example,
represent inequalities describing nutritional and cost constraints
on combinations of different foods
Solve real world application problems involving piecewise, step, and
absolute value functions
CCSS: A-CED.2,3
MACLE: AR.3.A
Solve a system of functions (on a calculator and by hand)
Verify the solution(s), and explain the meaning of the solutions (systems
may include linear, polynomial, rational, absolute value, exponential,
and logarithmic functions)
CCSS: A-REI.11
MACLE: AR.1.C
Compare properties of two functions presented in different formats
(Formats may include algebraically, graphically, table or verbal
descriptions)
CCSS: F-IF.9
MACLE: AR.2.C
Instructional Strategies
•
•
•
•
•
•
Lecture enhanced with:
• SMART Notebook
• PowerPoint
• the Internet
Drill and guided practice
Reflective discussion
Student self-reflection
Class discussion
Computer assisted instruction
Board Approved 7-15-13
Revised 2013
23
Algebra 2
Assessments/Evaluations
The students will be assessed on the concepts taught using a variety of modalities:
• Direct teacher observations
• Potential project with a scoring guide: http://illuminations.nctm.org/LessonDetail.aspx?id=L852
• Quizzes
• Practice/homework assignments – see pacing guide
• Formal common assessment – Unit F test
Mastery Level: 80%
Instructional Resources/Tools
•
•
•
Textbook(s): (sample copies on bookshelf)
• Pearson – Algebra 2 Common Core (primary source)
• Holt – Algebra 2 Common Core Edition
• Houghton Mifflin – On Core Mathematics: Algebra 2
Website(s): www.mathsisfun.com/sets/functions-piecewise.html
Graphing calculator
Board Approved 7-15-13
Revised 2013
24
Algebra 2
Conceptual Category(s)
Domain
Cluster
Alignments:
CCSS: See below
Performance: 1.6
Knowledge: (MA) 4
MACLE: See below
NETS: 1c; 2d; 4c
DOK: 1-3
•
•
•
•
•
•
•
•
•
•
•
•
Algebra
Functions
Creating Equations (A-CED)
Seeing Structures in Expressions (A-SSE)
Interpreting Functions (F-IF)
Building Functions (F-BF)
Linear, Quadratic, and Exponential Models (F-LE)
Create equations that describe numbers or relationships (A-CED.4)
Write expressions in equivalent forms to solve problems (A-SSE.4)
Understand the concept of a function and use function notation (F-IF.3)
Build a function that models a relationship between two quantities (F-BF.2)
Construct and compare linear, quadratic, and exponential models and solve problems (F-LE.2)
Standards
A-CED.4
4. Rearrange formulas to highlight a quantity of interest, using the
same reasoning as in solving equations. For example, rearrange
Ohm’s law V = IR to highlight resistance R
A-SSE.4
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.
For example, calculate mortgage payments ★
Learning Targets
Unit G: Sequences and Series: Use sequences and series to analyze and
solve problems
•
•
Solve formulas for a specified variable
Rearrange arithmetic and geometric sequences and series
CCSS: A-CED.4
MACLE: AR.2.B
•
•
Derive the formula for the sum of the finite geometric series
Calculate and express finite geometric series
Board Approved 7-15-13
Revised 2013
25
Algebra 2
F-IF.3
3. Recognize that sequences are functions, sometimes defined
recursively, whose domain is a subset of the integers. For
example, the Fibonacci sequence is defined recursively by f(0) =
f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1
•
F-BF.1a, 2
1. Write a function that describes a relationship between two
quantities ★
a. Determine an explicit expression, a recursive process, or steps
for calculation from a context
2. Write arithmetic and geometric sequences both recursively and
with an explicit formula, use them to model situations, and
translate between the two forms
•
•
•
F-LE.2
2. 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)
Recognize and solve real-world problems involving finite geometric
series and sum of a finite geometric series
CCSS: A-SSE.4
MACLE: AR.2.A
Write the formula/function of the sequence
Use the recursive formula to generate the next term of the sequence
Distinguish between explicit and recursive formulas
CCSS: F-IF.3
MACLE: AR.1.B
•
•
•
Write explicit and recursive arithmetic sequence formulas
Write explicit and recursive geometric sequence formulas
Translate between the recursive and explicit forms of geometric
sequences
CCSS: F-BF.2
MACLE: AR.1.B,C; AR.2.A
•
Solve real-world problems involving explicit and recursive arithmetic
and geometric sequences
CCSS: F-BF.1
MACLE: AR.1.B
•
•
•
Determine if a function is linear or exponential given a sequence, a
graph, a verbal description, or a table
Construct a function from an arithmetic sequence, a table of values, or a
description of the relationship
Describe the process used to construct the linear or exponential function
that passes through two given points
CCSS: F-LE.2
MACLE: AR.1.B,C; AR.2.A
Board Approved 7-15-13
Revised 2013
26
Algebra 2
Instructional Strategies
•
•
•
•
•
•
•
•
•
•
•
Direct and explicit instruction
Guided and independent practice
Infuse AFL tenets in instruction
“We learn” through integration of instructional technology such as:
• SMART Notebook
• PowerPoint
• Prezi
• the Internet
• discovery-type experience
Graphic organizer: Formulas for arithmetic/geometric sequences and series
Small-group instruction/facilitation
Summarizing/synthesizing concepts
Student self-reflection
Teacher self-reflection
Class discussion
Game: Grains of Rice on a Chess Board – The students will practice assessment-like questions on sequences and series. For example, a question
may be asked:
When you place rice on the chess board:
•
•
•
1 grain on the first square,
2 grains on the second square,
4 grains on the third, and so on...
... doubling the grains of rice on each square ...
Board Approved 7-15-13
Revised 2013
27
Algebra 2
Assessments/Evaluations
The students will be assessed on the concepts taught using a variety of modalities:
• Direct teacher observations
• Projects with scoring guides – A Reality Series Project:
• Students will work with a partner to research a real-world application of arithmetic and geometric sequences and series
• At the conclusion of their research, they will make a poster presentation of their findings and present it to the class
• Quizzes
• Homework assignments – see pacing guide
• Formal Common Assessment – Unit G test
Mastery Level: 80%
Instructional Resources/Tools
•
•
•
•
•
Textbook(s): (sample copies on bookshelf)
• Pearson – Algebra 2 Common Core (primary source)
• Holt – Algebra 2 Common Core Edition
• Houghton Mifflin – On Core Mathematics: Algebra 2
Website(s): www.khanacademy.org
Graphing calculator
Computer simulation software
Sequence and series apps
Board Approved 7-15-13
Revised 2013
28