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HACKETTSTOWN PUBLIC SCHOOLS
HACKETTSTOWN, NEW JERSEY
COLLEGE PREP AND HONORS
PRECALCULUS
GRADES 11-12
CURRICULUM GUIDE
FINAL DRAFT
June 2012
Mr. Robert Gratz, Superintendent
Ms. Diane Pittenger, Assistant Superintendent for Curriculum and Instruction
Mr. Roy Huchel, Mathematics Supervisor
Developed by:
Michelle DeFilippis
Christine Montalbano
This curriculum may be modified through varying techniques,
strategies and materials, as per an individual student’s
Individualized Education Plan (IEP).
Approved by the Hackettstown Board of Education
At the regular meeting held on
8/8/2012
and
Aligned with the Common Core State Standards 2010
Hackettstown School District
Mathematics
Algebra 2 – Unit 1 Algebra Review
Stage 1: Desired Results
Topic: Algebra Review
Common Core State Standards
N.CN.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
A.APR.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.
A.APR.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.
A.SSE.1a Interpret expressions that represent a quantity in terms of its context.
a. Interpret parts of an expression, such as terms, factors, and coefficients.
A.SSE.3a 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 zeros of the function it defines.
A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at
the previous step, starting from the assumption that the original equation has a solution. Construct a viable
argument to justify a solution method.
Essential Questions
1. How do the properties of real numbers help us
in simplifying expressions?
2. Why is it important to be able to simplify
algebraic expressions?
3. What does it mean to solve an equation?
Enduring Understandings
1.
Properties of real numbers and equality can be
used to transform an expressions or equation
into an equivalent, simpler form that is easier
to use.
2. The number of solutions to an equation varies
predictably, based on the type of equation.
3. Many real world mathematical problems can
be represented algebraically. These
representations can lead to algebraic solutions.
Knowledge and Skills:
1. Recognize and apply properties of real numbers including closure, commutative, associative, identity
and distributive properties.
2. Simplify expressions such as
3. Factor polynomials involving greatest common factor, difference of two perfect squares, and trinomials.
4. Solve linear equations with variables on both sides of the equation.
5. Solve quadratic equations by factoring and setting the expression equal to zero.
6. Use the Fundamental Theorem of Algebra which guarantees at most one unique answer to a linear
equation and at most 2 distinct answers to a quadratic equation.
Learning Expectations/Objectives
21st Century Life and Careers
Integration of Digital Tools
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
Assessment Methods:
Formative:
Summative:
Other Evidence and Student Self-Assessment: (Project Based Learning)
Stage 3: Learning Plan
For this unit consider how you will”
Engage Students
A
Provide evidence of Differentiated Instruction
B
Allow students to revise, rethink, refine, rethink
C
Time Allotment: (Minutes of instruction for
days)
Resources:
Student Materials:
Technology:
Teaching Materials:
Teaching Resources:
Hackettstown School District
Mathematics
Algebra 2 – Unit 2 Linear Relationships and Functions
Stage 1: Desired Results
Topic: Linear Relationships and Functions
Common Core State Standards
N.CN.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials
N.VM (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in
a network.
N.VM.7 (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are
doubled.
N.VM.8 (+) Add, subtract, and multiply matrices of appropriate dimensions.
N.VM.9 (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not
a commutative operation, but still satisfies the associative and distributive properties.
N.VM.10 (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication
similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if
the matrix has a multiplicative inverse.
N.VM.11 (+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to
produce another vector. Work with matrices as transformations of vectors.
N.VM.12 (+) Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the
determinant in terms of area.
A.CED.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.
A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph
equations on coordinate axes with labels and scales.
A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients
represented by letters.
A.REI.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.
A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of
linear equations in two variables.
A.REI.8 (+) Represent a system of linear equations as a single matrix equation in a vector variable.
A.REI.9 (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using
technology for matrices of dimension 3 × 3 or greater).
A.REI.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).
A.REI.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.
F.IF.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).
F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use
function notation in terms of a context
F.IF.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 periodicity.
F.IF.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 function.
F.IF.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.
F.IF.7a Graph functions expressed symbolically and show key features of the graph, by hand in simple cases
and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
F.IF.7b 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.
F.BF.1b 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
+F.BF.1c 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.
F.BF.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.
F.BF.4a 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.
+F.BF.4b Find inverse functions.
b. (+) Verify by composition that one function is the inverse of another.
+F.BF.4c Find inverse functions.
c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
F.LE.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).
F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context.
Essential Questions
Enduring Understandings
1. How can we use mathematical models to
1. A function is a relationship between variables
describe change over time?
in which each value of the independent
variable is associated with a unique value of
2. How do graphs help us to solve equations?
the dependent variable.
3. What is the difference between a relation and a
function?
2. Functions can be represented in a variety of
4. How is a function related to its inverse and
ways including graphs, tables, equations, and
how is it used?
words.
3. New functions can be made from other
functions by applying arithmetic operations or
by applying one function to the output of
another.
4. Linear equations are useful in describing many
real world situations that involve constant rate
of change.
5. The slopes of parallel and perpendicular lines
are closely related.
6. Systems of equations can be used to solve
equations that model real-life situations.
7. Using inverse functions, we can “reverse” the
equations to find the input to a function that
would yield a desired output.
Knowledge and Skills:
1. Determine domain and range of a function using its graph.
2. Use the Vertical Line test to determine if a relation is a function.
3. Use function notation.
4. Graph a linear equation using a table of values, intercepts, and using slope and y-intercept information.
5. Identify linear and non-linear equations.
6. Write linear equations in standard form.
7. Write an equation of a line given its slope and y-intercept.
8. Write an equation of a line given slope and one point.
9. Write the equation of a line given two points that lie on the line.
10. Write equations of parallel and perpendicular lines.
11. Understand and use function operations including addition, subtraction, and multiplication.
12. Determine the inverse of a linear equation by switching the x and y coordinates.
13. Recognize that the inverse of a function is the reflection of the original graph over the line y = x.
14. + Perform function composition.
15. + Use function composition to verify inverse functions.
16. Graph piecewise functions
17. Graph step functions.
18. Graph absolute value equations and make connections between the equation’s absolute value form and
its piecewise form.
19. Transform absolute value equations by replacing f(x) by f(x)+k, kf(x), f(kx), and f(x+k) for specific
values of k (both positive and negative)
20. Solve a system of equations in 2 variables graphically, by substitution method, and by linear
combinations.
21. + Represent data in a matrix.
22. + Multiply matrices by a scalar.
23. + Add, subtract, and multiply matrices.
24. + Find the determinant.
25. +Find the area of a triangle using determinants.
Learning Expectations/Objectives
21st Century Life and Careers
Integration of Digital Tools
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
Assessment Methods:
Formative:
Summative:
Other Evidence and Student Self-Assessment: (Project Based Learning)
Stage 3: Learning Plan
For this unit consider how you will”
Engage Students
A
Provide evidence of Differentiated Instruction
B
Allow students to revise, rethink, refine, rethink
C
Time Allotment: (Minutes of instruction for
days)
Resources:
Student Materials:
Technology:
Teaching Materials:
Teaching Resources:
Day 1:
Day 2:
Day 3:
Day 4:
Day 5
Day 6
Hackettstown School District
Mathematics
Algebra 2- Unit 3 – Quadratic Functions
Stage 1: Desired Results
Topic: Quadratic Functions
Common Core State Standards
N.CN.1 Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi
with a and b real.
N.CN.2 Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract,
and multiply complex
numbers.
+N.CN.3 (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex
numbers.
+N.CN.4 (+) Represent complex numbers on the complex plane in rectangular and polar form (including real
and imaginary numbers), and explain
why the rectangular and polar forms of a given complex number represent the same number.
N.CN.7 Solve quadratic equations with real coefficients that have complex solutions
+N.CN.8 (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x –
2i).
N.CN.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials
A.SSE.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).
A.SSE.3a 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 zeros of the function it defines.
A.SSE.3b Choose and produce an equivalent form of an expression to reveal and explain properties of the
quantity represented by the expression.★
b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it
defines.
A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph
equations on coordinate axes with labels
and scales
A.REI.4a Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic equation in x into an equation of the
form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
A.REI.4b Solve quadratic equations in one variable.
b. 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 ± bi for real numbers a and b.
A.REI.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).
A.REI.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.★
F.IF.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).
F.IF.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 periodicity.★
F.IF.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
function.★
F.IF.7a Graph functions expressed symbolically and show key features of the graph, by hand in simple cases
and using technology for more
complicated cases.★
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
F.IF.7c Graph functions expressed symbolically and show key features of the graph, by hand in simple cases
and using technology for more
complicated cases.★
c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end
behavior.
F.IF.8a Write a function defined by an expression in different but equivalent forms to reveal and explain
different properties of the function.
a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values,
and symmetry of the graph, and interpret these in terms of a context.
F.BF.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.
Essential Questions
Enduring Understandings
1. What are the characteristics that define all
1. Quadratic equations can be used to model
polynomial functions of the second degree?
many real world situations.
2. How are quadratic equations used to model
2. Quadratic equations can be solved using a
projectile motion?
variety of techniques including inspection of
3. What is an imaginary number?
graphs or tables, rearranging equations and
taking the square root of both sides of the
equation, by factoring and setting the factors
equal to zero, or by using the quadratic
formula. No matter which method you use to
solve the equation, you will arrive at the same
answer.
3. The imaginary number is no less “real” than
any other number we currently know. It is a
concept that was invented by man to describe a
particular situation.
Knowledge and Skills:
1. Graph a quadratic equation using axis of symmetry, vertex, y-intercept, and roots of the equation.
2. Use the Fundamental Theorem of Algebra which guarantees at most 2 distinct answers to a quadratic
equation.
3. Understand various transformations that can be performed on a quadratic equation and describe its
effects on its graph.
4. Factor higher order polynomials such as x4 – y4.
5. Review simplification of square roots.
6. Solve equations by taking the square root of both sides of the equations such as x2 = 49.
7. Use method of completing the square to rewrite expressions and to solve quadratic equations.
8. Demonstrate the derivation of the quadratic formula and use it to solve polynomials of the second degree
in simplest radical form and decimal form.
9. Use and relate properties of the nature of the roots using the discriminant to graphically describe a
quadratic function in terms of its placement in the coordinate plane.
10. Understand and use the imaginary unit, i to solve quadratic equations with real coefficients that have
complex solutions.
11. Perform basic operations (add, subtract, multiply) with complex numbers and simplify .
12. Apply knowledge of quadratics to solve projectile motion problems.
+13. Find the conjugate of a complex number to rationalize and find absolute value.
+14. Graph complex numbers on the complex plane in rectangular form.
+15. Factoring polynomials with complex numbers such as
.
Learning Expectations/Objectives
21st Century Life and Careers
Integration of Digital Tools
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
Assessment Methods:
Formative:
Summative:
Other Evidence and Student Self-Assessment: (Project Based Learning)
Stage 3: Learning Plan
For this unit consider how you will”
Engage Students
A
Provide evidence of Differentiated Instruction
B
Allow students to revise, rethink, refine, rethink
C
Time Allotment: (Minutes of instruction for
days)
Resources:
Student Materials:
Technology:
Teaching Materials:
Teaching Resources:
Day 1:
Day 2:
Hackettstown School District
Mathematics
Algebra 2- Unit 4 Polynomial Functions
Stage 1: Desired Results
Topic: Polynomial Functions
Common Core State Standards
N.CN.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph
equations on coordinate axes with labels
and scales.
A.REI.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).
A.REI.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.SSE.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).
A.APR.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.
A.APR.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).
A.APR.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.
A.APR.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 the more
complicated examples, a computer algebra system
F.IF.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).
F.IF.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 periodicity.★
F.IF.7c 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases
and using technology for more
complicated cases.★
c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end
behavior.
Essential Questions
Enduring Understandings
1. What information does the degree of a
polynomial tell you about its related graph?
1.
A function is a relationship between variables
in which each value of the input variable is
How is this related to the Fundament Theorem
of Algebra?
2. How are factors, zeros, roots, and x-intercepts
related for a given function?
3. How are factors and roots related?
associated with a unique value of the output
value.
2. Useful information about equations, including
solutions, can be found by analyzing graphs or
tables.
3. The number of solutions to an equation varies
predictably, based on the type of equation.
4. By studying “parent” graphs and understanding
their general behavior, we can describe the end
behavior and the graph of other polynomial
functions of the same degree.
Knowledge and Skills:
1. Define and recognize polynomial functions. Use parent graphs of y = xn to describe end behavior of the
graph.
2. Use the Fundamental Theorem of Algebra which guarantees at most n unique answers for a polynomial
of degree n.
3. Factor polynomials to find zeros of a function.
4. Describe intervals of increasing/decreasing and intervals of positive/negative for a polynomial function.
5. Sketch a polynomial function using parent graphs, degree, zeros and y-intercepts.
6. Use the graphing calculator to locate the relative maxima and/or relative minima of a polynomial
function.
7. Given a graph of a polynomial functions, determine its domain and range.
8. Use long division and the remainder theorem to rewrite functions as the sum of the dividend and its
remainder.
9. Explain why the x-coordinate 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).
10. Find the solutions approximately using technology to graph functions or to make a table of values.
Learning Expectations/Objectives
21st Century Life and Careers
Integration of Digital Tools
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
Assessment Methods:
Formative:
Summative:
Other Evidence and Student Self-Assessment: (Project Based Learning)
Stage 3: Learning Plan
For this unit consider how you will”
Engage Students
A
Provide evidence of Differentiated Instruction
B
Allow students to revise, rethink, refine, rethink
C
Time Allotment: (Minutes of instruction for
days)
Resources:
Student Materials:
Technology:
Teaching Materials:
Teaching Resources:
Hackettstown School District
Mathematics
Algebra 2 – Unit 5 Exponents and Radicals
Stage 1: Desired Results
Topic: Exponents and Radicals
Core Content Curriculum Number & Strands
N.RN.1 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. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to
hold, so (51/3)3 must equal 5.
N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
A.SSE.3C 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the
expression.★
c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be
rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
A.CED.2
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels
and scales.
A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
A.REI.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).
A.REI.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.★
F.IF.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 periodicity.★
F.IF.7b 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.
F.IF.7c Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more
complicated cases.★
c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior
F.LE.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).
F.LE.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.
F.LE.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.
F.LE.5
Interpret the parameters in a linear or exponential function in terms of a context.
Essential Questions
1. What is the meaning of a fractional exponent?
2. How are negative exponents related to
fractions?
3. What is the relationship between exponents
and logs?
Enduring Understandings
1. There are different ways to solve equations
using radicals all of which result in the same
answer.
2. A logarithm is another way to represent an
exponential equation.
3. Exponential and logarithmic functions are
inverses of each other and reflections of the
line y = x.
Knowledge and Skills:
1. Use and apply the laws of exponents to simplify exponential expressions.
2. Simplify expressions involving fractional, negative, and zero exponents.
3. Change base to solve exponential equation such as
.
4. Operations with radicals and simplifying radicals like
,
,2
,
5. Find smallest value of x for which
represents a real number (domain and range for radicals)
6.
Solve and graph radical equations and identify when roots are extraneous.
7.
Graph exponential and log functions such as y = 2x, y = 2-x, y = -2x
a. The graph of y = ax contains (0,1)
b. The graph of y = logax contains (1,0)
c. These graphs are reflections over the line y = x
8. Change from exponential to log form of an equation base 2, 10, e
9. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity
increasing linearly, quadratically, or as a polynomial function.
10. Explain why the x-coordinate 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, use technology to
graph functions, make tables of values, find successive approximations for cases of logarithmic and
exponential functions.
11. Fit an exponential function to data.
Learning Expectations/Objectives
21 Century Life and Careers
Integration of Digital Tools
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
Assessment Methods:
Formative:
Summative:
Other Evidence and Student Self-Assessment: (Project Based Learning)
Stage 3: Learning Plan
For this unit consider how you will”
Engage Students
A
Provide evidence of Differentiated Instruction
B
Allow students to revise, rethink, refine, rethink
C
Time Allotment: (Minutes of instruction for
days)
Resources:
Student Materials:
Technology:
Teaching Materials:
Teaching Resources:
st
Day 1:
Day 2:
Hackettstown School District
Mathematics
Algebra 2 –Unit 6 Sequences and Series
Stage 1: Desired Results
Topic: Sequences and Series
Common Core State Standards
A.SSE.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.★
A.CED.2
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels
and scales.
F.IF.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 ε 1.
F.BF.1a
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.
F.BF.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
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).
Essential Questions
1. What is the concept of recursion?
2. What is the difference between an arithmetic
and geometric sequence?
3. How can you find the sum of an arithmetic or
geometric sequence?
4. What is summation notation and how is it used
in mathematics?
5. What is meant by the concept of a limit in
mathematics?
Enduring Understandings
1.
2.
3.
4.
5.
6.
7.
8.
If the numbers in a list follow a pattern, variables
may be used to relate each number in the list to its
numerical position in the list.
In an arithmetic sequence, the difference between
any two consecutive terms is always the same
number. This number can be represented by a
variable.
In a geometric sequence, the ratio of any term
(after the first) to its preceding term is a constant
value, no matter what two terms are compared. A
geometric sequence can be built by multiplying
each term by the constant.
When two terms and the number of terms in a
finite arithmetic sequence are known, a formula
can be used to find the sum of the terms.
The sum of a finite geometric series can be found
using a formula. The first term, the number of
terms, and the common ratio must be know.
If an infinite geometric series has a common ratio
less than one, then you can find the sum of the
series using a formula.
Summation notation is used to express sums of
sequences.
Graphs and table of values can be used to
determine if a sequence has a limit.
Knowledge and Skills:
1. Graph sequences and determine if a limit exists.
2. Use subscripts and formulas for sequences.
3. Write and use formulas for sequences in which each term is found by using the preceding terms
(recursive formulas)
4. Identify sequences that have a common difference, a common ratio, or neither to classify a sequence as
5.
6.
7.
8.
9.
10.
arithmetic or geometric.
Write explicit and recursive formulas for given sequences.
Find the sum of a finite, non-geometric series.
Use a formula to find the sum of a finite geometric series
Find the sum of an infinite geometric series if it exists.
Understand and use summation notation.
Express a series using summation notation.
Learning Expectations/Objectives
21 Century Life and Careers
Integration of Digital Tools
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
Assessment Methods:
Formative:
Summative:
Other Evidence and Student Self-Assessment: (Project Based Learning)
Stage 3: Learning Plan
For this unit consider how you will”
Engage Students
A
Provide evidence of Differentiated Instruction
B
Allow students to revise, rethink, refine, rethink
C
Time Allotment: (Minutes of instruction for
days)
Resources:
Student Materials:
Technology:
Teaching Materials:
Teaching Resources:
st
Day 1:
Day 2:
Day 3:
Hackettstown School District
Mathematics
Algebra 2- Unit 7 Statistics
Stage 1: Desired Results
Topic: Statistics
Common Core State Standards
S.ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard
deviation) of two or more different data sets.
S.ID.3
Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points
(outliers).
S.ID.4
Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. 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.ID.5
Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data
(including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
S.ID.6a Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested
by the context. Emphasize linear, quadratic, and exponential models.
S.ID.6b Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
b. Informally assess the fit of a function by plotting and analyzing residuals.
S.ID.6c Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
c. Fit a linear function for a scatter plot that suggests a linear association.
S.ID.8
S.ID.9
Compute (using technology) and interpret the correlation coefficient of a linear fit.
Distinguish between correlation and causation.
Essential Questions
Enduring Understandings
1. How can you effectively evaluate information?
2. How can you use information to make decisions?
3. How are measures of central tendency different
from measures of central dispersion?
4. What is the significance of standard deviation and
how does it help me to interpret a data set?
5. What is the meaning of correlation in a
mathematical context?
1. Large data sets can be summarized easily in
graphical form, hence, the saying “a picture is
worth a thousand words”.
2. Data can be collected, organized, sorted,
represented, and analyzed in a variety of ways.
3. Measures of central tendency help to simplify the
data set by describing it with a “typical” data value
for the entire set.
4. Measures of central tendencies can give an
incomplete picture of a data set. Measures of
dispersion are important in understanding and
interpreting a data set in regards to its measures of
central tendencies.
5. Many common statistics such as human height,
weight, or blood pressure gathered from samples
in the natural world tend to have a normal
distribution.
Knowledge and Skills:
1. Create histograms and relative frequency histograms for a given data set to better understand the
distribution of the data set.
2. Identify different types of frequency distributions including uniform, mound, skewed, and bimodal
distributions.
3. Calculate mean, median, mode, range and standard deviation by hand.
4. Use mean, range, and standard deviation to compare data sets.
5. Use graphing calculators to quickly determine measures of central tendency and dispersion.
6. Determine probabilities of occurrence in a normal distribution utilizing the 68 – 95 – 99.7 % Rule of
Thumb.
7. Use scatter plots to determine if a linear or quadratic model suit a given situation.
8. Use graphing calculators to determine the line of best fit.
9. Use mathematical models of a real world situation to make estimates or predictions about future
occurrences.
10. Interpret correlation coefficient to determine the appropriateness of a given mathematical model for a
given data set.
Learning Expectations/Objectives
21st Century Life and Careers
Integration of Digital Tools
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
Assessment Methods:
Formative:
Summative:
Other Evidence and Student Self-Assessment: (Project Based Learning)
Stage 3: Learning Plan
For this unit consider how you will”
Engage Students
A
Provide evidence of Differentiated Instruction
B
Allow students to revise, rethink, refine, rethink
C
Time Allotment: (Minutes of instruction for
days)
Resources:
Student Materials:
Technology:
Teaching Materials:
Teaching Resources:
Day 1:
Day 2:
Day 3:
Hackettstown School District
Mathematics
Algebra 2- Unit 8 Conic Sections
Stage 1: Desired Results
Topic: Conic Sections
Common Core State Standards
A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels
and scales.
A.REI.7
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example,
find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.
A.REI.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.★
G.GPE.1
Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and
radius of a circle given by an equation.
G.GPE.2 Derive the equation of a parabola given a focus and directrix.
+G.GPE.3 Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is
constant.
Essential Questions
1. What is the intersection of a cone and a plane?
2. How are conics useful?
3. How do you represent a conic algebraically?
Enduring Understandings
1. Each point of a parabola is equidistant from the
focus and directrix.
2. A circle is the set of points that is equidistant from
a given point.
3. The intersection of a line and a parabola or a circle
can be found graphically and algebraically.
Knowledge and Skills:
1. Find the equation and graph of a parabola given the focus or directrix and the center.
2. Find the equation and graph of a circle given a center and radius.
3. Find the intersection of a line and a circle both algebraically and graphically.
4. Find the intersection of a line and a parabola both algebraically and graphically.
+5. Find the equation and graph an ellipse and a hyperbola.
Learning Expectations/Objectives
21st Century Life and Careers
Integration of Digital Tools
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
Assessment Methods:
Formative:
Summative:
Other Evidence and Student Self-Assessment: (Project Based Learning)
Stage 3: Learning Plan
For this unit consider how you will”
Engage Students
A
Provide evidence of Differentiated Instruction
B
Allow students to revise, rethink, refine, rethink
C
Time Allotment: (Minutes of instruction for
days)
Resources:
Student Materials:
Technology:
Teaching Materials:
Teaching Resources:
Day 1:
Day 2:
Day 3:
Day 4:
Day 5
Day 6
Hackettstown School District
Mathematics
Algebra 2- Unit 9 Trigonometric Functions
Stage 1: Desired Results
Topic: Trigonometric Functions
Common Core State Standards
A.REI.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
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).
F.IF.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 periodicity.★
F.IF.7E
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.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.
F.TF.1
F.TF.2
Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
F.TF.5
F.TF.8
Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.★
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.
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.
Essential Questions
1. How can you model periodic behavior?
2. What type of real world problems can be modeled
and solved using trigonometry?
3. How are the trigonometric functions related?
Enduring Understandings
1. Radian measure is related to the length of an arc.
2. The unit circle enables the extension of trig
functions to all real numbers.
3. By studying “parent” graphs and understanding
their general behavior, we can graph and interpret
other trigonometric functions.
Knowledge and Skills:
1. Relate the radian measure of the length of an arc subtended by an angle.
2. Convert between radian and degree measures of an angle.
3. Find the length of an arc given the radius and angle measure.
4. Derive the sine, cosine, and tangent values for axial angles and 30, 45, and 60.
5. Use
=1 to find sine or cosine when given one value.
6. Calculate trig ratios ex if tan x = ¾, find cos x if x is quadrant III
7. Graph parent trig functions for sin x and cosine x only. Identify where increasing, decreasing, positive
and negative. Find domain and range.
8. Determine amplitude, frequency, and midline and use these to sketch the graph
and the
related cosine function.
9. Graph y = tan x.
Learning Expectations/Objectives
21 Century Life and Careers
Integration of Digital Tools
Stage 2: Evidence of Understanding
st
Hackettstown Benchmarks:
Students will:
Assessment Methods:
Formative:
Summative:
Other Evidence and Student Self-Assessment: (Project Based Learning)
Stage 3: Learning Plan
For this unit consider how you will”
Engage Students
A
Provide evidence of Differentiated Instruction
B
Allow students to revise, rethink, refine, rethink
C
Time Allotment: (Minutes of instruction for
days)
Resources:
Student Materials:
Technology:
Teaching Materials:
Teaching Resources:
Day 1:
Day 2:
Day 3:
Day 4:
Day 5