Scientific Investigation - Hanover County Public Schools

Number and
Number Sense
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Probability &
Expressions
Equations and
Systems
Statistics
Inequalities
Relations &
Functions
Virginia Standard:
AII.1 The student will identify field properties, axioms of equality and inequality, and properties of order that are valid for
the set of real numbers and its subsets, complex numbers, and matrices.
Hanover Objective(s):
same
Related Standard(s):
Curriculum Framework:
Essential Understandings
• Complex numbers are organized into a hierarchy of
subsets with properties applicable to each subset.
•
Complex numbers are a superset of real numbers and,
as a system, contain solutions for equations that are
not solvable over the set of real numbers.
Resources:
•
Essential Knowledge and Skills
Identify examples of field properties: commutative,
associative, identity, inverse, and distributive.
•
Identify examples of axioms of equality: reflexive,
symmetric, transitive, substitution, addition, and
multiplication.
•
Identify examples of axioms of inequality and order:
trichotomy, transitive, addition, and multiplication.
•
Place the following sets of numbers in a hierarchy of subsets:
complex, pure imaginary, real, rational, irrational, integers,
whole, and natural.
•
Add and multiply matrices, and determine which field
properties hold.
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Textbook Correlation
Activities
“Properties of Real Numbers”, Max A. Sobel, Norbert Lerner,
Glencoe Algebra II
Chapter 1 p. 6-46
Evan M. Matetsky, Louis s. Cohen, Harper & Row Algebra Two,
“Expressions and Formulas”
Scribner Educational Publishers, New York, NY: 1985, p. 13-15.
“Properties of Real Numbers”
[Hereafter referred to as Sobel and number.]
“Solving Equations”
Norton Juster, Dot and Line, Random House, New York: 1977.
“Solving Absolute Value Equations”
(Now available in paperback: ISBN 0-394-73352-5)
“Solving Inequalities”
Using Matrices in Real-Life Settings, Roland E. Larson, Algebra
“Solving Compound and Absolute Value Inequalities”
2, D.C. Heath and Company, 1993, p. 176.
Chapter 2 p. 63-67
I Have/Who Has(Instructional Activity AII.1a.ACT)
“Linear Equations”
Patterns from Pascal’s Triangle (Instructional Activity
Chapter 5 p. 270-275
AII.1B.ACT).
“Complex Numbers”
Fibonacci Sequence (Instructional Activity AII.1C.ACT)
Worksheet Review WS 1 – AII.1B
Noteables
Equation WS 1- AII.1C
p. 1-21
Equation WS 2 – AII.1D
p. 30-32
Math Block Table-AII.1E
p. 116-118
Practice 1-3-AII.1F
Math Tool Box Skills Review-AII.1G
Properties – AII.1H
Lesson 3-3, Solving Equations – AII.1I
Lesson 3-3, Combining Like Terms – AII.1J
Lesson 3-4, Using the Distributive Property,
Simplifying Variable Expressions – AII.1K
Lesson 3-4, Using the Distributive Property,
Solving and Modeling Equations – AII.1L
Practice 3-3, Mixed Exercises – AII.1M
Practice 3-4, Example Exercises – AII.1N
Practice 4-7, Mixed Exercises, AII.1O
Lesson 8-7, More Multiplication Properties of
Exponents, Raising a Product to a Power –
AII.1P
Lesson 8-7, More Multiplication Properties of
Exponents, Raising a Power to a Power –
AII.1Q
Lesson 8-6, A Multiplication Property of
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Exponents, Multiplying Powers – AII.1R
Lesson 8-8, Division Properties of Exponents,
Dividing Powers with the Same Base – AII.1S
Lesson 8-8, Division Properties of Exponents,
Raising a Quotient to a Power – AII.1T
Powere Properties Chart- AII.1U
Properties Table – AII.1V
Properties Worksheet-AII.1V
Test Review – AII.1X
Textbook:
“Properties and Formulas of Advanced Algebra”, p. T660, p. 662663.
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Number and
Number Sense
Probability &
Statistics
Expressions
Equations and
Inequalities
Systems
Relations &
Functions
Virginia Standard:
AII.2 The student will add, subtract, multiply, divide, and simplify rational expressions, including complex fractions.
Hanover Objective(s): same
Curriculum Framework:
Essential Understandings
• Computational skills applicable to numerical fractions also
apply to rational expressions involving variables.
•
•
Essential Knowledge and Skills
Add, subtract, multiply, and divide rational expressions
whose denominators are monomials or polynomial
expressions in completely factored form.
Simplify a rational expression with common monomial or
binomial factors.
• Recognize a complex fraction, and simplify it as a quotient or
product of simple fractions.
Resources:
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Textbook Correlation
Activities
Glencoe Algebra II
Chapter 1 p. 11-27
“Properties of Real Numbers”
“Solving Equations”
Chapter 2 p. 81-86
“Modeling Real-World Data: Using Scatter Plots”
Chapter 5 p. 222-228
“Monomials”
Chapter 9 p. 472-484
“Multiplying and Dividing Rational Expressions”
“Adding and Subtracting Rational Expressions”
Noteables
p. 6-10
p. 38-40
p. 98-100
p. 203-208
Resources
“ * and / Rational Expressions”Ronald E. Larson, Timothy D. Kanold,
and Lee Stiff, Algebra 2, D. C. Heath and Company, Lesington, MA:
1993, p. 533-540 [Hereafter referred as Larson and page number]
“+ and – Rational Expressions” Larson, p. 548-550
Worksheets
Factoring Worksheet- AII.2A
Operations with Fractions- AII.2B
Rational Expressions-AII.2C
Assessment:
Simplifying Rational Expressions-AII.2A.ASMT
Multiply/Divide Rational Exp (3 Forms)- AII.2B.ASMT
Simplify Rational Exp with bonus-AII.2C.ASMT
Make Up Test – AII.2D.ASMT
Operations with Rational Exp- AII.2E.ASMT
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Number and
Number Sense
Probability &
Statistics
Expressions
Equations and
Inequalities
Systems
Relations &
Functions
Virginia Standard:
AII.3 The student will:
• add, subtract, multiply, divide, and simplify radical expressions containing positive rational numbers and variables and
expressions containing rational exponents
• write radical expressions as expressions containing rational exponents, and vice versa.
Hanover Objective(s):
same
Curriculum Framework:
Essential Understandings
The student will:
• add, subtract, multiply, divide, and simplify radical
expressions containing positive rational numbers and
variables and expressions containing rational exponents
• write radical expressions as expressions containing
rational exponents, and vice versa.
• simplify radical expressions containing numbers and
variables
• add radical expressions
• subtract radical expressions
• multiply radical expressions
• divide radical expressions
• write expressions with rational exponents in simplest
radical form and vice versa.
•
Essential Knowledge and Skills
Simplify radical expressions containing positive rational
numbers and variables.
•
Convert from radical notation to exponential notation, and
vice versa.
•
Add and subtract radical expressions with like radicands.
•
Multiply and divide radical expressions not requiring
rationalizing the denominators.
Resources:
Textbook Correlation
Activities
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Glencoe Algebra II
Chapter 5 p. 233-238, p. 245-262
“Dividing Polynomials”
“Roots of Real Numbers”
“Radical Expressions”
“Rational Exponents”
Noteables:
p. 103-104
p. 107-113
Resources:
Properties Of Roots of Real Numbers” Larson, p. 368-371 and Max A.
Sobel, Norbert Lerner, Evan M. Maletsky, and Louis S. Cohen, Algebra
2, Harper & Row, 1985, p. 174, 214 [Hereafter referred to as Sobel and
page number.]
Worksheets:
Radical WS 1 – AII.3A
Radical WS 2 – AII. 3B
Radical WS 3 – AII.3C
Radical WS 4 – AII.3D
Radical WS 5 – AII.3E
Lesson 9-4, Simplifying Radicals, Multiplications with Radicals –
AII.3F
Lesson 9-4, Simplifying Radicals, Division with Radicals – AII.3G
Lesson 9-5, Adding and Subtracting Radicals –
AII.3H
Reteaching 9-4 – AII.3I
Lesson 9-5, Adding and Subtracting Radicals –
AII.3J
Reteaching 9-5 – AII.3K
Assessment:
Radical Expressions/Equations – AII.3A.ASMT
Simplifying Radicals and Solving Equations (2 Forms) – AII.3B.ASMT
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Number and
Number Sense
Probability &
Statistics
Expressions
Equations and
Inequalities
Systems
Relations &
Functions
Virginia Standard:
AII.4 The student will solve absolute value equations and inequalities graphically and algebraically. Graphing calculators
will be used both as a primary method of solution and to verify algebraic solutions.
Hanover Objective(s): same
Curriculum Framework:
Essential Understandings
• Absolute value equations and inequalities can be used to
model practical problems.
Essential Knowledge and Skills
• Solve absolute value equations in one variable algebraically
and graphically, using a graphing calculator.
• Solve absolute value inequalities in one variable
algebraically and graphically.
• Express the solutions to absolute value equations and
inequalities in one variable graphically and as an algebraic
inequality.
• Graph absolute value equations in two variables.
•Verify solutions to absolute value equations and inequalities in
two variables, using a graphing calculator.
Resources:
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Textbook Correlation
Activities
Glencoe Algebra II
Resources
Chapter 1 p. 40-46
“Solving Absolute Value Equations and Inequalities” Larson, p.
“Solving Compound and Absolute Value Inequalities”
44-46.
Chapter 9 p. 499-504
Worksheets:
“Classes of Functions”
Practice 4-8, Absolute Value Inequalities – AII.4A
Noteables:
Assessment Sample:
p. 16-18
Sobel, p. 78 and p. A8.
p. 213-214
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Number and
Number Sense
Probability &
Statistics
Expressions
Equations and
Inequalities
Systems
Relations &
Functions
Virginia Standard:
AII.5 The student will identify and factor completely polynomials representing the difference of squares, perfect square
trinomials, the sum and difference of cubes, and general trinomials.
Hanover Objective(s): same
Curriculum Framework:
Essential Understandings
• The complete factorization of polynomials has occurred
when each factor is a prime polynomial.
Essential Knowledge and Skills
• Determine the greatest monomial factor as a first step in
complete factorization.
• Pattern recognition can be used to determine complete
factorization of a polynomial.
• Recognize squares and cubes of positive integers.
•
Recognize examples of general patterns: difference of
squares, sum and difference of cubes, and perfect square
trinomials.
• Factor polynomials by applying general patterns.
Resources:
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Textbook Correlation
Activities
Glencoe Algebra II
Chapter 5 p. 239-244
“Factoring Polynomials”
Chapter 6 p. 294-305
“Solving Quadratic Equations by Graphing”
“Solving Quadratic Equations by Factoring”
Noteables
p. 105-106
p. 129-133
Resources:
“Factoring Polynomials and Solving Polynomial Equations” Larson, p.
473-475.
Worksheets:
Practice 10-2- AII.5A
Practice 10-4, Example Exercises- AII.5B
Practice 10-5, Mixed Exercises – AII.5C
Practice 10-6, Example Exercises – AII.5D
Practice 10-6, Mixed Exercises – AII.5E
10-2,Multiplying and Factoring with Monomials10-4,Factoring Trinomials using Tiles – AII.5G
10-4, Factoring Trinomials – AII.5H
10-5, Factoring Special Cases – AII.5I
10-6, Solve Equations by Factoring – AII.5J
AII.5F
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Number and
Number Sense
Probability &
Statistics
Expressions
Equations and
Inequalities
Systems
Relations &
Functions
Virginia Standard:
AII.6 The student will select, justify, and apply a technique to solve a quadratic equation over the set of complex numbers.
Graphing calculators will be used for solving and confirming algebraic solutions.
Hanover Objective(s): same
Related Standard(s):
Curriculum Framework:
Essential Understandings
• A quadratic equation whose graph does not intersect the xaxis has only complex solutions.
• Complex solutions occur in pairs (conjugates).
• The quadratic formula can be used to solve any quadratic
equation.
Resources:
Essential Knowledge and Skills
• Recognize a quadratic equation.
• Select an appropriate strategy for solving a quadratic
equation (factoring, using the quadratic formula, or
graphing).
•
Solve a quadratic equation over the set of complex
numbers.
•
Identify from a graph the real solutions to a quadratic
equation.
•
Find the real roots of a quadratic equation, using a
graphing calculator.
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Textbook Correlation
Activities
Glencoe Algebra II
Worksheets:
Chapter 6 p. 294-321
Equation WS 1 – AII.6A
“Solving Quadratic Equations by Graphing”
Equation WS 2 – AII.6B
“Solving Quadratic Equations by Factoring”
“Completing the Square”
“The Quadratic Formula and the Discriminant”
Chapter 9 p. 499-504
“Classes of Functions”
Noteables
p. 129-139
p. 213-214
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Number and
Number Sense
Probability &
Statistics
Expressions
Equations and
Inequalities
Systems
Relations &
Functions
Virginia Standard:
AII. 7 The student will solve equations containing rational expressions and equations containing radical expressions
algebraically and graphically. Graphing calculators will be used for solving and confirming algebraic solutions.
Hanover Objective(s): same
Curriculum Framework:
Essential Understandings
• A solution of an equation makes the equation true.
• Equations can be solved in a variety of ways.
Essential Knowledge and Skills
• Solve equations containing rational expressions with
monomial denominators algebraically and graphically.
• The solution of an equation in one variable can be found
by graphing each side of the equation separately and
finding the x-coordinate of the point of intersection.
• Solve equations containing a radical expression
algebraically and graphically. The equation will contain
a linear expression under the radical, and all terms
outside the radical will be constants.
• Practical problems can be interpreted, represented, and
solved using equations.
•
Identify from a graph the solutions to an equation
containing rational or radical expressions.
•
Solve an equation containing rational or radical
expressions, using a graphing calculator.
•
Check possible solutions to an equation containing
rational or radical expressions, using a graphing
calculator.
Resources:
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Textbook Correlation
Activities
Glencoe Algebra II
Chapter 1 p.20-27
“Solving Equations”
Chapter 5 p. 263-267
“Radical Equations and Inequalities”
Chapter 9 p. 499-511
“Classes of Functions”
“Solving Rational Equations and Inequalities”
Noteables
p. 8-10
p. 114-115
p. 213-218
Resources:
Exploring Rational Functions (Instructional
Activity AII.7A).
“Solving Radical Equations” , Larson p. 374-378. “Solving Rational
Equations” , Larson p. 541-551.
Worksheets:
Exploring Rational Functions (2 Pages)-AII.7A
9-6,Solving Radical Equations-AII.7B
9-6, Solving Radical Equations w/extraneous solutions- AII.7C
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Number and
Number Sense
Probability &
Statistics
Expressions
Equations and
Inequalities
Systems
Relations &
Functions
Virginia Standard:
AII.8 The student will recognize multiple representations of functions (linear, quadratic, absolute value, step, and
exponential functions) and convert between a graph, a table, and symbolic form. A transformational approach to
graphing will be employed through the use of graphing calculators.
Hanover Objective(s): same
Curriculum Framework:
Essential Understandings
• The graphs/equations for a family of functions can be
determined using a transformational approach.
Essential Knowledge and Skills
• Recognize graphs of parent functions for linear, quadratic,
absolute value, step, and exponential functions.
• Given an equation of a function, identify the function as
linear, quadratic, absolute value, step, or exponential.
• Write the equation of a linear (slope-intercept form),
quadratic ([h, k] form), absolute value, step, or
exponential function, given the graph of the parent
function or an integral translation of a parent function.
• Given an equation, graph a linear, quadratic, absolute value,
step, or exponential function with the aid of a graphing
calculator.
Resources:
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Textbook Correlation
Activities
Glencoe Algebra II
Worksheets:
Chapter 2 p. 56-80, 89-99
Piecewise Functions (Activity AII.8A)
“Relations and Functions”
Postage and the Greatest Integer Function
“Linear Equations’
(Instructional Activity AII.8B)
“Slope”
Doubling Time Growth Model (Activity AII.8C)
“Writing Linear Equations”
Investment Banking (Activity AII.8D)
“Special Functions”
Algebra II Group Activity (Activity AII.8E)
“Graphing Inequalities”
Rational Functions – AII.8F
Chapter 5 p. 229-232
Graphing Step, Absolute Value and Linear
“Polynomials”
Functions – AII.8G
Chapter 6 p. 286-299, 322-328
Building A Bridge (2 Pages) AII.8H
“Graphing Quadratic Functions”
Relationships and Graphs (Packet) AII.8I
“Solving Quadratic Equations by Graphing”
“Analyzing Graphs of Quadratic Functions”
Assessment:
Graphing Calculator Investigation p. 320-321
AII Quiz-Lesson 1.7, 1.3 (4 Forms) AII.8A.ASMT
Chapter 7 p. 346-352
Test on Review of Alg I and Chap 1 (4 Forms)
“Polynomial Functions”
AII.8B.ASMT
Chapter 8 p. 419-425
Make up Test, Review of Alg I and Chap 1 (2
“Parabolas”
AII.8C.ASMT
Chapter 9 p. 499-504
Quiz 2.1, 2.3- AII.8D.ASMT
“Classes of Functions”
Test 2 (1-5) and Lesson 8.1- AII.8E.ASMT
Chapter 10 p. 523-530
“Exponential Functions”
Noteables
p. 26-37
p. 41-44
p. 101-102
p. 126-131
p. 140-141
p. 152-154
p. 181-183
p. 213-214
p. 226-228
Forms)
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Number and
Number Sense
Probability &
Statistics
Expressions
Equations and
Inequalities
Systems
Relations &
Functions
Virginia Standard:
AII.9 The student will find the domain, range, zeros and inverse of a function, the value of a function for a given element in
its domain, and the composition of multiple functions. Functions will include exponential, logarithmic, and those that
have domains and ranges that are limited those that have domains and ranges that are limited and/or discontinuous.
The Graphing calculator will be used as a tool to assist in investigation of functions, including exponential and
logarithmic.
Hanover Objectives: same
Curriculum Framework:
Essential Understandings
• Functions describe the relationship between two
variables.
Essential Knowledge and Skills
• Identify the domain, range, zeros, and inverse of a function
presented algebraically or graphically.
• Graphs of functions that are inverses of each other are
reflections across the line y = x.
• Distinguish between relations and functions that are
expressed algebraically and graphically.
• The composition of a function and its inverse is the
identity function.
• Recognize restricted/discontinuous domains and ranges.
• Functions arise from practical situations.
• If (a, b) is an element of a function, then (b, a) is an
element of the inverse of the function.
•
Use interchange of variables to find the inverse of a
function.
•
Given the graphs, recognize that exponential and
logarithmic functions are inverses of each other.
•
Find the composition of two functions.
•
Find the value of a function for a given element from the
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
domain.
•
Investigate exponential and logarithmic functions, using
the graphing calculator.
Resources:
Textbook Correlation
Glencoe Algebra II
Chapter 2 p. 56-62
“Relations and Functions”
Chapter 6 p. 286-293
“Graphing Quadratic Functions”
Chapter 7 p. 383-394
“Operations on Functions”
“Inverse Functions and Relations”
Chapter 9 p. 499-504
“Classes of Functions”
Noteables
p. 26-29
p. 126-128
p. 167-170
p. 213-214
Activities
Worksheets:
Domain and Range (Instructional Activity AII.9A)
WS – Section 5-4 – AII.9B
WS – Section 5-4 – AII.9C
WS – Section 5-4 – AII.9D
Relations and Functions (Packet) AII.9E
Assessment Sample:
Restrict your teaching to graphing logs on the calculator (example
y = log x
y = log 2x
y= 2 log x.)
Teach how to use the “ log button” on the calculator.
Emphasize domain and range on every graphing problem.
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Number and
Number Sense
Probability &
Statistics
Expressions
Equations and
Inequalities
Systems
Relations &
Functions
Virginia Standard:
AII.10
The student will investigate and describe the relationships between the solution of an equation, zero of a
function, x-intercept of a graph, and factors of a polynomial expression through the use of graphs.
Hanover objectives: same
Curriculum Framework:
Essential Understandings
Essential Knowledge and Skills
• The Fundamental Theorem of Algebra states that,
including complex and repeated solutions, an nth degree
polynomial equation has exactly n roots (solutions).
•
Identify the x-intercept(s) of a graph.
•
Identify the zero(s) of a function, given a graph.
• The following statements are equivalent:
k is a zero of the polynomial function f;
− (x – k) is a factor of f(x);
− k is a solution of the polynomial equation f(x) = 0;
and
− k is an x-intercept for the graph of the polynomial.
•
Determine the linear factors of a polynomial expression
when the zeros of the corresponding polynomial function
are displayed on a graph.
Resources:
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Textbook Correlation
Activities
Glencoe Algebra II
Worksheets:
Chapter 2 p. 63-80
Relationships and Graphs (Packet) AII.10A
“Linear Equations”
“Slope”
“Writing Linear Equations”
Chapter 5 p. 239-244
“Factoring Polynomials”
Chapter 6 p. 294-305
“Solving Quadratic Equations by Graphing”
“Solving Quadratic Equations by Factoring”
Chapter 7 p. 346-352
“Polynomial Functions”
Noteables
p. 30-37
p. 105-106
p. 129-133
p. 152-154
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Number and
Number Sense
Probability &
Statistics
Expressions
Equations and
Inequalities
Systems
Relations &
Functions
Virginia Standard:
AII. 11 The student will use matrix multiplication to solve practical problems. Graphing calculators or computer
programs with matrix capabilities will be used to find the product.
Hanover Objective(s): same
Curriculum Framework:
Essential Understandings
• Matrices can be used to model and solve practical
problems.
Essential Knowledge and Skills
• Recognize matrices that can be multiplied.
• Perform matrix multiplication with a graphing calculator
or a computer program with matrix capabilities.
•
Resources:
Use matrix multiplication to solve practical problems.
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Textbook Correlation
Activities
Glencoe Algebra II
Worksheets:
Chapter 4 p. 154-174, 182-188, 195-207
Assignment Sheet for Chapter 3 (2 pages) AII.11A
“Introduction to Matrices”
“Operations with Matrices”
Assessment:
“Multiplying Matrices”
Test 3.1-3.3, 3.6 (2 Forms)-No Calculator-AII.11A.ASMT
“Determinants”
Quiz 3.1-3.3 AII.11B.ASMT
“Identity and Inverse Matrices”
Test Chap 3 (w/calc) AII.11C.ASMT
“Using Matrices to Solve Systems of Equations”
Make Up Test 3 (w/calc) AII.11D.ASMT
Noteables
p. 72-79
p. 83-84
p. 87-90
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Number and
Number Sense
Probability &
Statistics
Expression
Equations and
Inequalities
Systems
Relations &
Functions
Virginia Standard:
AII.12 The student will represent problem situations with a system of linear equations and solve the system using the
inverse matrix method. Graphing calculators of computer programs with matrix capability will be used to perform
computations.
Hanover Objective(s): same
Curriculum Framework:
Essential Understandings
• Matrices are a convenient shorthand for solving systems
of equations.
Essential Knowledge and Skills
• Model problems with a system of no more than three
linear equations.
• Matrices can model a variety of linear systems.
• Represent a system of no more than three linear equations
in matrix form.
• Solutions of a linear system are values that satisfy every
equation in the system.
Resources:
• Solve a matrix equation using a graphing calculator or
computer program with matrix capability.
•
Find the inverse of a matrix with a graphing calculator.
•
Express a system of linear equations as a matrix
equation.
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Textbook Correlation
Activities
Glencoe Algebra II
Chapter 3 p. 110-122
“Solving Systems of Equations by Graphing”
“Solving Systems of Equations Algebraically”
Noteables
p. 52-57
“Linear Systems”, Larson, p. 207-208.
Systems of Equations & Inequalities – AII.12A
Matrices for Problems in Manufacturing –
AII.12B
Using Matrices to Organize Data – AII.12C
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Number and
Number Sense
Probability &
Statistics
Expression
Equations and
Inequalities
Systems
Relations &
Functions
Virginia Standard:
AII.13
The student will solve systems of linear inequalities and linear programming problems and describe the results both
orally and in writing. A graphing calculator will be used to facilitate solutions to linear programming problems.
Hanover Objective(s): same
Curriculum Framework:
Essential Understandings
• Linear programming models an optimization process.
• A linear programming model consists of a system of
constraints and an objective quantity that can be
maximized or minimized.
• Any maximum or minimum value will occur at a corner
point of a feasible region.
Resources:
Essential Knowledge and Skills
• Model practical problems with systems of linear
inequalities.
• Solve systems of linear inequalities.
• Identify the feasibility region of a system of linear
inequalities with no more than five constraints.
•
Identify the coordinates of the corner points of a
feasibility region.
•
Find the maximum or minimum value for the function
defined over the feasibility region.
•
Describe the meaning of the maximum or minimum
value.
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Textbook Correlation
Activities
Algebra II Activity (Instructional Activity
Glencoe Algebra II
Chapter 3 p. 123-135
AII.13A)
“Solving Systems of Inequalities by Graphing”
Linear Programming – AII.13B
“Linear Programming”
Graphing Systems – AII.13C
Noteables
p. 58-63
UCSMP:
Systems of Linear Inequalities, Section 5.7
Linear Programming I, Section 5.8
Linear Programming II, Section 5.9
“Concepts of Linear Programming”, Bellman, p. 300-304, 313,
561.
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Number and
Number Sense
Probability &
Statistics
Expression
Equations and
Inequalities
Systems
Relations &
Functions
Virginia Standard:
AII. 14
The student will solve nonlinear systems of equations, including linear-quadratic and quadratic-quadratic,
algebraically and graphically. The graphing calculator will be used as a tool to visualize graphs and predict the
number of solutions.
Hanover Objective(s): same
Curriculum Framework:
Essential Understandings
• Solutions of a nonlinear system of equations are numerical
values that satisfy every equation in the system.
Essential Knowledge and Skills
• Identify nonlinear systems of equations as linear-quadratic
or quadratic-quadratic.
• The coordinates of points of intersection in nonlinear
systems are solutions to the system.
• Visualize a nonlinear system of two equations, and predict
the number of solutions, using the graphing calculator.
Resources:
•
Solve a linear-quadratic system of two equations
algebraically and graphically.
•
Solve a quadratic-quadratic system of two equations
algebraically and graphically.
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Textbook Correlation
Activities
“Nonlinear Systems” , Sobel, p. 348-349, 352-353.
Glencoe Algebra II
Chapter 8 p. 455-460
Systems – AII.14A
“Solving Quadratic Systems”
UCSMP:
Noteables
Quadratic-Linear Systems, Section 12.9
p. 194-196
Quadratic-Quadratic Systems, Section 12.10
“Linear-Quadratic Systems”, Sobel, p. 346-349.
“Quadratic Systems”, Sobel, p. 350-353.
“Systems with Nonlinear Equations”, Bellman, p. 305-308.
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Number and
Number Sense
Probability &
Statistics
Expression
Equations and
Inequalities
Systems
Relations &
Functions
Virginia Standard:
AII. 15
The student will recognize the general shape of polynomial, exponential, and logarithmic functions. The graphing
calculator will be used as a tool to investigate the shape and behavior of polynomial functions.
Hanover Objective(s): same
Curriculum Framework:
Essential Understandings
• Shapes and behavior of graphs of polynomials can be
determined by analyzing transformations of parent
functions.
Essential Knowledge and Skills
• Investigate the shape and behavior of linear, quadratic, and
cubic functions. Behaviors will include intercepts,
number of turning points, and end behavior.
• Using graphing calculators is a strategy for investigating
the shape and behavior of polynomial functions.
•
Investigate the shape and behavior of exponential (ax =
y) and logarithmic (log b x = y) functions, including
intercepts and end behavior.
•
Using the general shape of the graph of a function,
identify the family of graphs to which a particular graph
belongs. Characteristics of a graph may include the xand y-intercepts, number and location of turning points,
and end behaviors.
• The Fundamental Theorem of Algebra (Carl Fredrich
Gauss) states that in the complex number system, an nth
degree polynomial equation has n zeros.
• Exponential and logarithmic functions are either strictly
increasing or strictly decreasing.
Resources:
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Textbook Correlation
Activities
Circle (Instructional Activity AII.15A)
Glencoe Algebra II
Chapter 6 p. 286-293, 322-328
Ellipses (Instructional Activity AII.15B)
“Graphing Quadratic Functions”
Hyperbolas (Instructional Activity AII.15C)
“Analyzing Graphs of Quadratic Functions”
Parabolas (Instructional Activity AII.15D)
Chapter 9 p. 499-504
Projectile Motion (Instructional Activity AII.15E)
“Classes of Functions”
“Identifying Conics” , Kenneth J. Travers, Leroy C. Dalton and
Chapter 10 p. 523-538
Vincent F. Brunner, Using Advanced Algebra, Laidlaw Brothers,
River Forest, Illinois, 1975, p. 335.
“Exponential Functions”
“Logarithms and Logarithmic Functions”
UCSMP:
Noteables
Polynomial Models, Section 11.1
p. 126-128
Functions, Chapter 7
p. 140-141
“The Factor Theorem”, Section 11.4
p. 213-214
“Quadratic-Linear Systems”, Section 12.9
p. 226-230
“Quadratic-Quadratic Systems”, Section 12.10
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Number and
Number Sense
Probability &
Statistics
Expression
Equations and
Inequalities
Systems
Relations &
Functions
Virginia Standard:
AII. 16 The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve
problems, including writing the first n terms, finding the nth term, and evaluating summation formulas. Notation
will include Σ and an.
Hanover Objective(s): same
Curriculum Framework:
Essential Understandings
• Sequences and series arise from practical situations.
• The study of sequences and series is an application of
investigation of patterns.
Essential Knowledge and Skills
• Distinguish between a sequence and a series.
• Recognize patterns in a sequence.
• Distinguish between arithmetic and geometric sequences.
• Use and interpret the notations ∑, n, nth term, and an.
• Write the first n terms in an arithmetic or geometric
sequence.
• Given the formula, find an (the nth term) for an arithmetic
or a geometric sequence.
•
Resources:
Given formulas, find the sum, Sn, of the first n terms of
an arithmetic or geometric series, including infinite
series.
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Textbook Correlation
Activities
Seating Capacity, Roland E. Larson, Timothy
Glencoe Algebra II
Chapter 11 p. 578-598
D. Kanold, and Lee Stiff, Algebra 2, D.C.
“Arithmetic Sequences”
Heath and Company, Lesington, MA: 1993,
“Arithmetic Series”
p. 632.
“Geometric Sequences”
Sharing Ideas Larson, p. 640.
“Geometric Series”
Sharing Ideas Larson, p. 641.
Zooming Inn Larson, p. 649.
Noteables
p. 248-256
UCSMP:
Explicit Formulas for Sequences, Section 1-3
Recursive Formulas for Sequences, Section 1-4
Arithmetic Sequences: Explicit Formulas,
Section 3.6
Arithmetic Sequences: Recursive Formulas
Section 3.7
Geometric Sequences, Section 8.3
Series, Combinations, and Statistics, Sections 13.1-13.4
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Number and
Number Sense
Probability &
Statistics
Expression
Equations and
Inequalities
Systems
Relations &
Functions
Virginia Standard:
AII.17 The student will perform operations on complex numbers and express the results in simplest form. Simplifying
results will involve using patterns of the powers of i.
Hanover Objective(s): same
Curriculum Framework:
Essential Understandings
• Complex numbers are a superset of real numbers.
Essential Knowledge and Skills
• Recognize that the square root of –1 is represented as i.
• Define and identify a complex number.
• Apply the definition of i to simplify square roots of
negative numbers.
• Simplify powers of i.
•
Textbook Correlation
Glencoe Algebra II
Chapter 5 p. 270-275
“Complex Numbers”
Noteables
p. 116-118
Add, subtract, and multiply complex numbers.
Activities
Simplifying Expressions Containing Complex
Numbers – AII.17A
UCSMP:
The Imaginary Number i, Section 6..8
Complex Numbers, Section 6.9
“Complex Numbers”, Sobel, p. 296-300.
Number and
Number Sense
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Probability &
Expression
Equations and
Systems
Statistics
Inequalities
Relations &
Functions
Virginia Standard:
AII.18 The student will identify conic sections (circle, ellipse, parablola, and hyperbola) from his/her equations. Given the
equations in (h, k) form, students will sketch graphs of conic sections, using transformations.
Hanover Objective(s): same
Curriculum Framework:
Essential Understandings
• Changing parameters (h, k) affects the graph of a conic
section in a predictable pattern.
• Conic sections can be distinguished by their equations.
Essential Knowledge and Skills
• Identify types of conic sections, given (h, k) form of an
equation.
•
Identify types of conic sections from a graph.
•
Sketch the graph of a conic section in (h, k) form, using
knowledge of transformations.
Resources:
Textbook Correlation
Glencoe Algebra II
Chapter 8 p. 419-452
“Parabolas”’
“Circles”
“Ellipses”
“Hyperbolas”
“Conic Sections”
Noteables
p. 181-193
Activities
Golden Gate Bridge, Ronald E. Larson, Timothy D. Kanold, and
Lee Stiff, Algebra 2, D.C.
Heath and Company, Lesington, MA: 1993,
p.241.
Sleepy Hare Restaurant Larson, p. 275
Gateway Arch Larson, p. 249.
Circles, [Instructional Activity AII.15A]
Ellipses, [Instructional Activity AII.15B]
Hyperbolas, [Instructional Activity AII.15C]
Parabolas, [Instructional Activity AII.15D]
UCSMP:
Parabolas and Quadratic Equations, Chapter 6 (omit 6.3)
Quadratic Relations, Chapter 12, Sections 12.1 – 12.8
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Number and
Number Sense
Probability &
Statistics
Expression
Equations and
Inequalities
Systems
Relations &
Functions
Virginia Standard:
AII.19 The student will collect and analyze data to make predictions, write equations, and solve practical problems.
Graphing calculators will be used to investigate scatter plots to determine the equation for a curve of best fit. Models
will include linear, quadratic, exponential, and logarithmic functions.
Hanover Objective(s): same
Curriculum Framework:
Essential Understandings
• Data and scatterplots may indicate patterns that can be
modeled with an algebraic equation.
• Graphing calculators can be used to collect, organize,
picture, and create an algebraic model of the data.
• Data that fit linear, quadratic, exponential, and logarithmic
models arise from practical situations.
Essential Knowledge and Skills
• Collect and analyze data.
• Investigate scatterplots to determine if patterns exist, and
then identify the patterns.
• Find an equation for the curve of best fit for data, using a
graphing calculator. Models will include linear,
quadratic, exponential, and logarithmic functions.
•
Make predictions, using data, scatterplots, or curve of
best fit.
•
Given a set of data, determine the model that would best
describe the data.
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Resources:
Textbook Correlation
Glencoe Algebra II
Chapter 2 p. 87
Graphing Calculator Investigation
Chapter 6 p. 300
Graphing Calculator Investigation
Chapter 7 p. 359
Graphing Calculator Investigation
Essential Knowledge and Skills
Pulse Rate and Child’s Age (Instructional
Activity AII.19A)
“Scatter Plots”, Rubenstein, p. 39-40
“Scatter Plots II” , Rubenstein, p. 13.
TI-82/83 Procedure 23: Fining Best-Fit Models –
AII.19B
Scatter plot – AII.19C
Scatter plot and Line of Best Fit – AII.19D
UCSMP:
Fitting a Model to Data I, Section 2.8
Fitting a Model to Data II, Section 2.9
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Number and
Number Sense
Probability &
Statistics
Expression
Equations and
Inequalities
Systems
Relations &
Functions
Virginia Standard:
AII.20 The student will identify, create, and solve practical problems involving inverse variation and a combination of
direct and inverse variations.
Hanover Objective(s): same
Curriculum Framework:
Essential Understandings
• Practical problems can be modeled and solved by using
direct and/or inverse variations.
• Joint variation is a combination of direct variations.
Essential Knowledge and Skills
• Translate “y is directly proportional to x” as y = kx.
k
• Translate “y is inversely proportional to x” as y = x .
• Translate “y varies jointly as x and z” as y = kxz.
• Determine the value of the constant of proportionality, k,
given initial conditions for x and y.
•
Set up and solve practical problems, using combinations
of direct and inverse variation.
MATHEMATICS
Mathematics Curriculum---Hanover County Public Schools
Revised Summer 2005
Resources:
Textbook Correlation
Glencoe Algebra II
Chapter 9 p. 492-504
“Direct, Joint, and Inverse Variation”
“Classes of Functions”
Noteables:
p. 211-214
Activities
Distance to a Thunderstorm (Instructional
Activity AII.20A)
Temperature Determines Chirps (Instructional
Activity AII.20B)
Direct, Inverse, and Joint Variations – AII.20C
Direct, Inverse, and Joint Variations – AII.20D
Direct Variation – AII.20E
Inverse Variation – AII.20F
Variation Review – AII.20G
UCSMP:
Variation and Graphs, Chapter 2.1-2.6

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