# Culpeper County Public Schools by 9WvuL0lX

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Curriculum Pacing Guide

High School Semester Algebra I
Revised June 2011
A YEAR AT A GLANCE ALGEBRA I
High School Semester Algebra I
(June 2011)
Benchmark 1                            Benchmark 2                        Benchmark 3                   Benchmark 4
UNIT 1 (1.5 weeks)                 UNIT 3 (1 week)                      UNIT 5 (2.5 weeks)             UNIT 7 (3 days)

Problem Solving - Strategies Functions –                                Exponent (A.2 a)                Best Fit Curves (A.11)
(A.4 f)                      *Analyze linear and quadratic
function families                 Polynomial Operations          SOL PREDICTOR TEST
Expressions (A.1)                           (A.7 a-f)                   (A.2 b)                        (13 weeks)

Multi-Step Equations (A.4          Direct and Inverse Variation         Factoring                      UNIT 8 (1.5 weeks)
d)                                      (A.8)                           (A.2 c)
Statistics – Analyzing Data Sets
Properties (A.4 b)                 UNIT 4 (2 weeks)                     UNIT 6 (1.5 weeks))               (Box and Whisker Plots)
(A.10)
Multi-Step Inequalities with       Systems of Linear Equations          Radicals (A.3)
one variable (A.5 a-c)             (A.4 e-f) – (Include Real World                                     Deviations and Z -Scores (A.9)
Literal Equations (A.4 a)                                               (Review A.7 with Quadratics)
Systems of Linear Inequalities
(A.5 d)                                                            SOL REVIEW
UNIT 2 (2.5 weeks)
COUNTY BENCHMARK II
Linear Equations in Two            ( 8 weeks)
Variables (A.4 d)

Slopes and Lines (A.6 a)
Equations of Lines (A.6 b)
Problem Solving with
Linear
Equations/Inequalities
(A.4 f)
COUNTY BENCHMARK I
(4.5 weeks)
CCPS Curriculum Guide                          Course Map                                                June 2011
THIS COURSE: Algebra I
is
about             the study of the rules of operations and relations, and the constructions and concepts arising
from them, including terms, polynomials, equations and algebraic structures

1.   How are algebraic expressions and equations used to solve real world problems?
2.   What is the relationship between a line and its graph, slope, intercepts, and table of values?
3.   How is a function, in all its forms, used to investigate relationships between quantitative data?
4.   How is a linear system used to solve real world problems?
5.   What are the different ways to simplify a polynomial expression?
6.   What is the most effective way to solve a given quadratic equation?
7.   How is a line or curve of best fit used to make predictions about a set of data?
8.   How are measures of deviation and z-scores used to describe and analyze a set a data?
CCPS Curriculum Guide                               Course Map                                         June 2011

Critical Vocabulary and/or Concepts

Solve                      Linear Equations              Box and Whisker Plots         Properties
Quadratic Equations        Systems                       Polynomials                   Rate of Change
Absolute Value             Inequalities                  Zeros                         Factoring
Simplify                   Exponents                     Domain and Range              Absolute Deviation
Linear Inequalities        Problem Solving               Variations                    Z-Scores
Slope                      Reciprocal                    Functions

Learned in
Basic Algebra
these
Concepts                                                UNITS
Statistics
A.4 b,d,f                                                                                      A.9 and A.10
A.1
A.5 a-c

Slopes and                                                                                Best Fit
Lines                                                                                  Curves
A.4 d,f                                                                                  A.11
A.6

A.7 and A.8                                                      A.3, A.4 c

Systems
Polynomials
A.4 e,f
A.2
A.5 d
Benchmark I
Unit 1
Standard A.4 a, b, d, f
Subject/Grade Level: Algebra I               Suggested Pacing for Benchmarking
Period: UNIT 1
Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including
a) solving literal equations (formulas) for a given variable;
b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that
are valid for the set of real numbers and its subsets;
c) solving quadratic equations algebraically and graphically;
d) solving multistep linear equations algebraically and graphically;
e) solving systems of two linear equations in two variables algebraically and graphically; and
f) solving real-world problems involving equations and systems of equations.
Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions.
Essential Knowledge                Resources and                Suggested
Essential Understandings
and Skills                      Activities             Assessment(s)
The student will use problem solving,
    A solution to an equation is the         mathematical communication,
Unit 1 Organizer         Interactive Achievement
value or set of values that can be       mathematical reasoning, connections,                               Unit Tests, Benchmark Tests,
substituted to make the equation                                                   Introductory
and representations to                                             and Projects
true.                                                                              Function Problems
   Solve a literal equation (formula)     (see Instr. Specialist
    The solution of an equation in one          for a specified variable.              Sue Jenkins)
variable can be found by graphing
the expression on each side of the         Simplify expressions and solve         Cube Train (see Toni
equation separately and finding the         equations, using the field             Miller)
x-coordinate of the point of                properties of the real numbers and
intersection.                               properties of equality to justify      Algeblocks or
simplification and solution.           Algebra Tiles
    Real-world problems can be
interpreted, represented, and solved       Solve quadratic equations.             Hands-On Equations
equations.                                 Identify the roots or zeros of a       Property Match
quadratic function over the real       Game (see Toni
    The process of solving linear and           number system as the solution(s)       Miller)
in a variety of ways, using concrete,       formed by setting the given            Cover-Up Problem
pictorial, and symbolic                     quadratic expression equal to zero.    (VDOE ESS (2004)
representations.
Subject/Grade Level: Algebra I             Suggested Pacing for Benchmarking
Period: UNIT 1
Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including
a) solving literal equations (formulas) for a given variable;
b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that
are valid for the set of real numbers and its subsets;
c) solving quadratic equations algebraically and graphically;
d) solving multistep linear equations algebraically and graphically;
e) solving systems of two linear equations in two variables algebraically and graphically; and
f) solving real-world problems involving equations and systems of equations.
Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions.
Essential Knowledge               Resources and                Suggested
Essential Understandings
and Skills                     Activities             Assessment(s)
 Properties of real numbers and               Solve multistep linear equations in
properties of equality can be used to         one variable.                         Algeblocks and
justify equation solutions and                                                      Equation Solving
expression simplification.                   Confirm algebraic solutions to        VDOE ESS (2004)
using a graphing calculator.
 The zeros or the x-intercepts of the                                                A Mystery to Solve
quadratic function are the real              Given a system of two linear          VDOE ESS (2004)
root(s) or solution(s) of the                 equations in two variables that has
quadratic equation that is formed by          a unique solution, solve the system   Solving Linear
setting the given quadratic                   by substitution or elimination to     Equations VDOE
expression equal to zero.                     find the ordered pair which           ESS (2004)
satisfies both equations.
 A system of linear equations with                                                   Video Power of
exactly one solution is characterized        Given a system of two linear          Algebra
by the graphs of two lines whose              equations in two variables that has
intersection is a single point, and           a unique solution, solve the system   Algebra/ Solving
the coordinates of this point satisfy         graphically by identifying the        Equations Jeopardy
both equations.                               point of intersection.                Style
 A system of two linear equations                                                    (http://quia.com)
 Determine whether a system of
with no solution is characterized by          two linear equations has one
the graphs of two lines that are                                                    Rags to Riches
solution, no solution, or infinite    (http://quia.com )
parallel.
Subject/Grade Level: Algebra I              Suggested Pacing for Benchmarking
Period: UNIT 1
Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including
a) solving literal equations (formulas) for a given variable;
b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that
are valid for the set of real numbers and its subsets;
c) solving quadratic equations algebraically and graphically;
d) solving multistep linear equations algebraically and graphically;
e) solving systems of two linear equations in two variables algebraically and graphically; and
f) solving real-world problems involving equations and systems of equations.
Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions.
Essential Knowledge               Resources and                Suggested
Essential Understandings
and Skills                      Activities             Assessment(s)
 A system of two linear equations              solutions.
having infinite solutions is
characterized by two graphs that             Write a system of two linear
coincide (the graphs will appear to           equations that models a real-world    Algebra Tiles
be the graph line), and the                   situation.                            (NCTM website)
coordinates of all points on the line                                               http://illuminations.n
satisfy both equations.                      Interpret and determine the           ctm.org/activitydetail.
reasonableness of the algebraic or    aspx?id=216 )
 Systems of two linear equations can           graphical solution of a system of
be used to model two real-world               two linear equations that models a
conditions that must be satisfied             real-world situation.                 Promethean Board Site
simultaneously.
 Determine if a linear equation in
 Equations and systems of equations            one variable has one, an infinite
can be used as mathematical models            number, or no solutions.†
for real-world situations.

   Set builder notation may be used to
represent solution sets of equations.

†
Revised March 2011
Benchmark I
Unit 1
Standard A.1
Subject/Grade Level: Algebra I               Suggested Pacing for Benchmarking
Period: UNIT 1
Standard: A.1 The student will represent verbal quantitative situations algebraically and evaluate these expressions for given
replacement values of the variables.
Essential Knowledge                   Resources and               Suggested
Essential Understandings
and Skills                         Activities            Assessment(s)
The student will use problem solving,
    Algebra is a tool for reasoning         mathematical communication,
Unit 1 Organizer
Interactive Achievement
about quantitative situations so that   mathematical reasoning, connections,     Algeblocks or Algebra
relationships become apparent.          and representations to                   Tiles                    Unit Tests, Benchmark Tests,
and Projects
    Algebra is a tool for describing and                                             Algebra Magic
   Translate verbal quantitative
representing patterns and                                                        Packet (see Inst.
situations into algebraic
relationships.                                                                   Specialist Sue
expressions and vice versa.
Jenkins)
    Mathematical modeling involves
   Model real-world situations with
creating algebraic representations of                                            Traffic Jam VDOE
algebraic expressions in a variety
quantitative real-world situations.                                              ESS (2004)
of representations (concrete,
pictorial, symbolic, verbal).
    The numerical value of an                                                        Expression Bingo
expression is dependent upon the
   Evaluate algebraic expressions for   (see Toni Miller)
values of the replacement set for the
a given replacement set to include
variables.                                                                       Evaluating and
rational numbers.
Simplifying
    There are a variety of ways to
   Evaluate expressions that contain    Expressions VDOE
compute the value of a numerical                                                 ESS (2004)
absolute value, square roots, and
expression and evaluate an
cube roots.
algebraic expression.
Promethean Board Site
    The operations and the magnitude
of the numbers in an expression
impact the choice of an appropriate
computational technique.

    An appropriate computational
technique could be mental
mathematics, calculator, or paper
Subject/Grade Level: Algebra I           Suggested Pacing for Benchmarking
Period: UNIT 1
Standard: A.1 The student will represent verbal quantitative situations algebraically and evaluate these expressions for given
replacement values of the variables.
Essential Knowledge                   Resources and               Suggested
Essential Understandings
and Skills                         Activities            Assessment(s)
and pencil.
Benchmark I
Unit 1
Standard A.5 a-c
Subject/Grade Level: Algebra I                Suggested Pacing for Benchmarking
Period: UNIT 1
Standard: A.5 The student will solve multistep linear inequalities in two variables, including
a) solving multistep linear inequalities algebraically and graphically;
b) justifying steps used in solving inequalities, using axioms of inequality and properties of order that are valid for the set
of real numbers and its subsets;
c) solving real-world problems involving inequalities; and
d) solving systems of inequalities.
Essential Knowledge                  Resources and                Suggested
Essential Understandings
and Skills                       Activities             Assessment(s)
The student will use problem solving,
    A solution to an inequality is the        mathematical communication,
Unit 1 Organizer        Interactive Achievement
value or set of values that can be        mathematical reasoning, connections,
substituted to make the inequality                                                  Inequalities VDOE
and representations to                    ESS (2004)
true.
   Solve multistep linear inequalities
    Real-world problems can be
in one variable.                       Promethean Board Site
modeled and solved using linear
inequalities.                               Justify steps used in solving
inequalities, using axioms of
    Properties of inequality and order
inequality and properties of order
can be used to solve inequalities.
that are valid for the set of real
numbers.
    Set builder notation may be used to
represent solution sets of
   Solve real-world problems
inequalities.
involving inequalities.

   Solve systems of linear
inequalities algebraically and
graphically.
Benchmark I
Unit 2
Standard A.4 d, f
Subject/Grade Level: Algebra I             Suggested Pacing for Benchmarking
Period: UNIT 2
Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including
a) solving literal equations (formulas) for a given variable;
b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that
are valid for the set of real numbers and its subsets;
c) solving quadratic equations algebraically and graphically;
d) solving multistep linear equations algebraically and graphically;
e) solving systems of two linear equations in two variables algebraically and graphically; and
f) solving real-world problems involving equations and systems of equations.
Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions.
Essential Knowledge                Resources and                 Suggested
Essential Understandings
and Skills                      Activities            Assessment(s)
Unit 2 Organizer
The student will use problem solving,
 A solution to an equation is the             mathematical communication,
Interactive Achievement
value or set of values that can be                                                    Greetings VDOE
mathematical reasoning, connections,     ESS (2004)
substituted to make the equation             and representations to
true.
 Solve a literal equation (formula) for  Algeblocks: Solving
 The solution of an equation in one             a specified variable.                  for Y VDOE ESS
variable can be found by graphing                                                     (2004)
the expression on each side of the           Simplify expressions and solve
equation separately and finding the            equations, using the field properties  Estimating Ages of
x-coordinate of the point of                   of the real numbers and properties of  Famous People
intersection.                                  equality to justify simplification and (mathbits.com)
solution.
 Real-world problems can be
interpreted, represented, and solved         Solve quadratic equations.              Promethean Board Site
equations.                                   Identify the roots or zeros of a
 The process of solving linear and              number system as the solution(s) to
in a variety of ways, using concrete,          by setting the given quadratic
pictorial, and symbolic                        expression equal to zero.
representations.
 Solve multistep linear equations in
 Properties of real numbers and
Subject/Grade Level: Algebra I             Suggested Pacing for Benchmarking
Period: UNIT 2
Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including
a) solving literal equations (formulas) for a given variable;
b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that
are valid for the set of real numbers and its subsets;
c) solving quadratic equations algebraically and graphically;
d) solving multistep linear equations algebraically and graphically;
e) solving systems of two linear equations in two variables algebraically and graphically; and
f) solving real-world problems involving equations and systems of equations.
Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions.
Essential Knowledge               Resources and                Suggested
Essential Understandings
and Skills                     Activities             Assessment(s)
properties of equality can be used to         one variable.
justify equation solutions and
expression simplification.                   Confirm algebraic solutions to
using a graphing calculator.
 The zeros or the x-intercepts of the
quadratic function are the real              Given a system of two linear
root(s) or solution(s) of the                 equations in two variables that has
quadratic equation that is formed by          a unique solution, solve the system
setting the given quadratic                   by substitution or elimination to
expression equal to zero.                     find the ordered pair which
satisfies both equations.
 A system of linear equations with
exactly one solution is characterized        Given a system of two linear
by the graphs of two lines whose              equations in two variables that has
intersection is a single point, and           a unique solution, solve the system
the coordinates of this point satisfy         graphically by identifying the
both equations.                               point of intersection.
   A system of two linear equations          Determine whether a system of
with no solution is characterized by       two linear equations has one
the graphs of two lines that are           solution, no solution, or infinite
parallel.                                  solutions.
   A system of two linear equations
Subject/Grade Level: Algebra I              Suggested Pacing for Benchmarking
Period: UNIT 2
Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including
a) solving literal equations (formulas) for a given variable;
b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that
are valid for the set of real numbers and its subsets;
c) solving quadratic equations algebraically and graphically;
d) solving multistep linear equations algebraically and graphically;
e) solving systems of two linear equations in two variables algebraically and graphically; and
f) solving real-world problems involving equations and systems of equations.
Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions.
Essential Knowledge               Resources and                Suggested
Essential Understandings
and Skills                      Activities             Assessment(s)
having infinite solutions is                 Write a system of two linear
characterized by two graphs that              equations that models a real-world
coincide (the graphs will appear to           situation.
be the graph of one line), and the
coordinates of all points on the line        Interpret and determine the
satisfy both equations.                       reasonableness of the algebraic or
graphical solution of a system of
 Systems of two linear equations can           two linear equations that models a
be used to model two real-world               real-world situation.
conditions that must be satisfied
simultaneously.                              Determine if a linear equation in
one variable has one, an infinite
number, or no solutions.†
   Equations and systems of equations
can be used as mathematical models
for real-world situations.

   Set builder notation may be used to
represent solution sets of equations.
†
Revised March 2011
Benchmark I
Unit 2
Standard A.6
Subject/Grade Level: Algebra I             Suggested Pacing for Benchmarking
Period: UNIT 2
Standard: A.6 The student will graph linear equations and linear inequalities in two variables, including
a) determining the slope of a line when given an equation of the line, the graph of the line, or two points on the line. Slope
will be described as rate of change and will be positive, negative, zero, or undefined; and
b) writing the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the
line.
Essential Knowledge                    Resources and               Suggested
Essential Understandings
and Skills                         Activities             Assessment(s)
The student will use problem solving,
   Changes in slope may be described       mathematical communication,
Unit 2 Organizer       Interactive Achievement
by dilations or reflections or both.    mathematical reasoning, connections,      Slippery Slope
and representations to                    VDOE ESS (2004)
   Changes in the y-intercept may be
described by translations.
   Graph linear equations and             Sally Snail VDOE
inequalities in two variables,
   Linear equations can be graphed                                                   ESS (2004)
including those that arise from a
using slope, x- and y-intercepts,
variety of real-world situations.      The Submarine
and/or transformations of the parent
function.                                                                         VDOE ESS (2004)
   Use the parent function y = x and
describe transformations defined
   The slope of a line represents a                                                  Transformationally
constant rate of change in the             by changes in the slope or y-          Speaking VDOE
dependent variable when the                intercept.                             ESS (2004)
independent variable changes by a
   Find the slope of the line, given      Transformation
constant amount.
the equation of a linear function.     Investigation) VDOE
   The equation of a line defines the                                                ESS (2004)
   Find the slope of a line, given the
relationship between two variables.
coordinates of two points on the       Slope 2 Slope VDOE
line.
   The graph of a line represents the                                                ESS (2004)
set of points that satisfies the
   Find the slope of a line, given the
equation of a line.                                                               Equations of Lines
graph of a line.
VDOE ESS (2004)
   A line can be represented by its
   Recognize and describe a line with
graph or by an equation.
a slope that is positive, negative,
   The graph of the solutions of a
Subject/Grade Level: Algebra I              Suggested Pacing for Benchmarking
Period: UNIT 2
Standard: A.6 The student will graph linear equations and linear inequalities in two variables, including
a) determining the slope of a line when given an equation of the line, the graph of the line, or two points on the line. Slope
will be described as rate of change and will be positive, negative, zero, or undefined; and
b) writing the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the
line.
Essential Knowledge                   Resources and               Suggested
Essential Understandings
and Skills                         Activities            Assessment(s)
linear inequality is a half-plane             zero, or undefined.                      Equations of Attack
bounded by the graph of its related                                                    (NCTM
linear equation. Points on the              Use transformational graphing to          http://illuminations.n
boundary are included unless it is a          investigate effects of changes in        ctm.org/lessondetail.a
strict inequality.                            equation parameters on the graph         spx?id=L782 )
of the equation.
 Parallel lines have equal slopes.                                                       Movie Lines (NCTM
 Write an equation of a line when          http://illuminations.n
 The product of the slopes of                   given the graph of a line.               ctm.org/lessondetail.a
perpendicular lines is -1 unless one                                                   spx?id=L629 )
of the lines has an undefined slope.
 Write an equation of a line when          The Line Runner
given two points on the line whose       (NCTM
coordinates are integers.                http://illuminations.n
ctm.org/lessondetail.a
 Write an equation of a line when          spx?id=L851 )
given the slope and a point on the
line whose coordinates are               School has this
integers.                                software : Green
Globs and Graphing
 Write an equation of a vertical line      Equations (very
as x = a.                                good)
   Write the equation of a horizontal
line as y = c.                          Promethean Board Site
Benchmark II
Unit 3
Standard A.7
Subject/Grade Level: Algebra I                Suggested Pacing for Benchmarking
Period: UNIT 3
Standard: A.7 The student will investigate and analyze function (linear and quadratic) families and their characteristics both
algebraically and graphically, including
a) determining whether a relation is a function;
b) domain and range;
c) zeros of a function;
d) x- and y-intercepts;
e) finding the values of a function for elements in its domain; and
f) making connections between and among multiple representations of functions including concrete, verbal, numeric,
graphic, and algebraic.
Essential Knowledge           Resources and                Suggested
Essential Understandings
and Skills                   Activities             Assessment(s)
The student will use problem solving,
    A set of data may be characterized       mathematical communication,
Unit 3 Organizer         Interactive Achievement
by patterns, and those patterns can      mathematical reasoning, connections,
be represented in multiple ways.                                                   Functions VDOE
and representations to                    ESS (2004)
    Graphs can be used as visual
   Determine whether a relation,         Square Patio VDOE
representations to investigate
represented by a set of ordered       ESS (2004)
relationships between quantitative
pairs, a table, or a graph is a
data.                                                                              Functions 2 VDOE
function.
ESS (2004)
    Inductive reasoning may be used to
   Identify the domain, range, zeros,
and intercepts of a function
characteristics of function families.                                              Representation
presented algebraically or
graphically.                          (NCTM
    Each element in the domain of a                                                    http://illuminations.n
relation is the abscissa of a point of
   For each x in the domain of f, find   ctm.org/lessondetail.a
the graph of the relation.                                                         spx?id=L621 )
f(x).
    Each element in the range of a                                                     Roller Coasting
   Represent relations and functions
relation is the ordinate of a point of                                             through Functions
using concrete, verbal, numeric,
the graph of the relation.                                                         (NCTM
graphic, and algebraic forms.
Given one representation, students    http://illuminations.n
    A relation is a function if and only                                               ctm.org/lessondetail.a
will be able to represent the
if each element in the domain is
Subject/Grade Level: Algebra I                Suggested Pacing for Benchmarking
Period: UNIT 3
Standard: A.7 The student will investigate and analyze function (linear and quadratic) families and their characteristics both
algebraically and graphically, including
a) determining whether a relation is a function;
b) domain and range;
c) zeros of a function;
d) x- and y-intercepts;
e) finding the values of a function for elements in its domain; and
f) making connections between and among multiple representations of functions including concrete, verbal, numeric,
graphic, and algebraic.
Essential Knowledge            Resources and              Suggested
Essential Understandings
and Skills                 Activities            Assessment(s)
paired with a unique element of the          relation in another form.          spx?id=L839 )
range.
 Detect patterns in data and
 The values of f(x) are the ordinates          represent arithmetic and geometric
of the points of the graph of f.             patterns algebraically.            Promethean Board Site

   The object f(x) is the unique object
in the range of the function f that is
associated with the object x in the
domain of f.

   For each x in the domain of f, x is a
member of the input of the function
f, f(x) is a member of the output of f,
and the ordered pair [x, f(x)] is a
member of f.

   An object x in the domain of f is an
x-intercept or a zero of a function f
if and only if f(x) = 0.

   Set builder notation may be used to
represent domain and range of a
relation.
Benchmark II
Unit 3
Standard A.8
Subject/Grade Level: Algebra I                Suggested Pacing for Benchmarking
Period: UNIT 3
Standard: A.8 The student, given a situation in a real-world context, will analyze a relation to determine whether a direct or
inverse variation exists, and represent a direct variation algebraically and graphically and an inverse variation algebraically.
Essential Knowledge                 Resources and               Suggested
Essential Understandings
and Skills                       Activities             Assessment(s)
The student will use problem solving,
    The constant of proportionality in a     mathematical communication,
Unit 3 Organizer         Interactive Achievement
direct variation is represented by the   mathematical reasoning, connections,
ratio of the dependent variable to                                                    Direct Variation
and representations to                       VDOE ESS (2004)
the independent variable.
   Given a situation, including a real-
    The constant of proportionality in                                                    Do I have to Mow the
world situation, determine whether       Whole Thing?
an inverse variation is represented
a direct variation exists.               (NCTM
by the product of the dependent
variable and the independent                                                          http://illuminations.n
   Given a situation, including a real-     ctm.org/lessondetail.a
variable.                                    world situation, determine whether       spx?id=L729 )
an inverse variation exists.
    A direct variation can be
represented by a line passing
   Write an equation for a direct           Promethean Board Site
through the origin.                          variation, given a set of data.
    Real-world problems may be                  Write an equation for an inverse
modeled using direct and/or inverse          variation, given a set of data.
variations.
   Graph an equation representing a
direct variation, given a set of data.
Benchmark II
Unit 4
Standard A.4 e, f
Subject/Grade Level: Algebra I               Suggested Pacing for Benchmarking
Period: UNIT 4
Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including
a) solving literal equations (formulas) for a given variable;
b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that
are valid for the set of real numbers and its subsets;
c) solving quadratic equations algebraically and graphically;
d) solving multistep linear equations algebraically and graphically;
e) solving systems of two linear equations in two variables algebraically and graphically; and
f) solving real-world problems involving equations and systems of equations.
Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions.
Essential Knowledge                Resources and                Suggested
Essential Understandings
and Skills                      Activities             Assessment(s)
The student will use problem solving,
    A solution to an equation is the         mathematical communication,
Unit 4 Organizer         Interactive Achievement
value or set of values that can be       mathematical reasoning, connections,
substituted to make the equation                                                   Road Trip VDOE
and representations to                    ESS (2004)
true.
   Solve a literal equation (formula)
    The solution of an equation in one          for a specified variable.              What’s Your Call?
variable can be found by graphing                                                  VDOE ESS (2004)
the expression on each side of the         Simplify expressions and solve
equation separately and finding the         equations, using the field             Spring Fling VDOE
x-coordinate of the point of                properties of the real numbers and     ESS (2004)
intersection.                               properties of equality to justify
simplification and solution.           The Exercise Ring
    Real-world problems can be                                                         VDOE ESS (2004)
interpreted, represented, and solved       Solve quadratic equations.
using linear and quadratic                                                         Talk or Text (NCTM
equations.                                 Identify the roots or zeros of a       Equations of Attack
quadratic function over the real       (NCTM
    The process of solving linear and           number system as the solution(s)       http://illuminations.n
quadratic equations can be modeled          to the quadratic equation that is      ctm.org/lessondetail.a
in a variety of ways, using concrete,       formed by setting the given            spx?id=L780 )
pictorial, and symbolic                     quadratic expression equal to zero.
representations.
   Solve multistep linear equations in
    Properties of real numbers and
Subject/Grade Level: Algebra I             Suggested Pacing for Benchmarking
Period: UNIT 4
Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including
a) solving literal equations (formulas) for a given variable;
b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that
are valid for the set of real numbers and its subsets;
c) solving quadratic equations algebraically and graphically;
d) solving multistep linear equations algebraically and graphically;
e) solving systems of two linear equations in two variables algebraically and graphically; and
f) solving real-world problems involving equations and systems of equations.
Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions.
Essential Knowledge               Resources and                Suggested
Essential Understandings
and Skills                     Activities             Assessment(s)
properties of equality can be used to         one variable.                         There Has to be a
justify equation solutions and                                                      System for this Sweet
expression simplification.                   Confirm algebraic solutions to        Problem (NCTM
using a graphing calculator.          ctm.org/lessondetail.a
 The zeros or the x-intercepts of the                                                spx?id=L766 )
quadratic function are the real              Given a system of two linear
root(s) or solution(s) of the                 equations in two variables that has
quadratic equation that is formed by          a unique solution, solve the system   Promethean Board Site
setting the given quadratic                   by substitution or elimination to
expression equal to zero.                     find the ordered pair which
satisfies both equations.
 A system of linear equations with
exactly one solution is characterized        Given a system of two linear
by the graphs of two lines whose              equations in two variables that has
intersection is a single point, and           a unique solution, solve the system
the coordinates of this point satisfy         graphically by identifying the
both equations.                               point of intersection.
   A system of two linear equations          Determine whether a system of
with no solution is characterized by       two linear equations has one
the graphs of two lines that are           solution, no solution, or infinite
parallel.                                  solutions.
   A system of two linear equations
Subject/Grade Level: Algebra I              Suggested Pacing for Benchmarking
Period: UNIT 4
Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including
a) solving literal equations (formulas) for a given variable;
b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that
are valid for the set of real numbers and its subsets;
c) solving quadratic equations algebraically and graphically;
d) solving multistep linear equations algebraically and graphically;
e) solving systems of two linear equations in two variables algebraically and graphically; and
f) solving real-world problems involving equations and systems of equations.
Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions.
Essential Knowledge               Resources and                Suggested
Essential Understandings
and Skills                      Activities             Assessment(s)
having infinite solutions is                 Write a system of two linear
characterized by two graphs that              equations that models a real-world
coincide (the graphs will appear to           situation.
be the graph of one line), and the
coordinates of all points on the line        Interpret and determine the
satisfy both equations.                       reasonableness of the algebraic or
graphical solution of a system of
 Systems of two linear equations can           two linear equations that models a
be used to model two real-world               real-world situation.
conditions that must be satisfied
simultaneously.                              Determine if a linear equation in
one variable has one, an infinite
number, or no solutions.
   Equations and systems of equations
can be used as mathematical models
for real-world situations.

   Set builder notation may be used to
represent solution sets of equations.
†
Revised March 2011
Benchmark II
Unit 4
Standard a,d
Subject/Grade Level: Algebra I                Suggested Pacing for Benchmarking
Period: UNIT 4
Standard: A.5 The student will solve multistep linear inequalities in two variables, including
a) solving multistep linear inequalities algebraically and graphically;
b) justifying steps used in solving inequalities, using axioms of inequality and properties of order that are valid for the set
of real numbers and its subsets;
c) solving real-world problems involving inequalities; and
d) solving systems of inequalities.
Essential Knowledge                  Resources and                Suggested
Essential Understandings
and Skills                       Activities             Assessment(s)
The student will use problem solving,
    A solution to an inequality is the        mathematical communication,
Unit 4 Organizer        Interactive Achievement
value or set of values that can be        mathematical reasoning, connections,
substituted to make the inequality        and representations to                    Promethean Board Site
true.
   Solve multistep linear inequalities
    Real-world problems can be
in one variable.
modeled and solved using linear
inequalities.                               Justify steps used in solving
inequalities, using axioms of
    Properties of inequality and order
inequality and properties of order
can be used to solve inequalities.
that are valid for the set of real
numbers.
    Set builder notation may be used to
represent solution sets of
   Solve real-world problems
inequalities.
involving inequalities.

   Solve systems of linear
inequalities algebraically and
graphically.
Predictor Test
Unit 5
Standard A.2
Subject/Grade Level: Algebra I              Suggested Pacing for Benchmarking
Period: UNIT 5
Standard: A.2 The student will perform operations on polynomials, including
a) applying the laws of exponents to perform operations on expressions;
b) adding, subtracting, multiplying, and dividing polynomials; and
c) factoring completely first- and second-degree binomials and trinomials in one or two variables. Graphing calculators
will be used as a tool for factoring and for confirming algebraic factorizations.
Essential Knowledge                 Resources and            Suggested
Essential Understandings
and Skills                       Activities          Assessment(s)
The student will use problem solving,
    The laws of exponents can be            mathematical communication,
Unit 5 Organizer         Interactive Achievement
investigated using inductive            mathematical reasoning, connections,     A.2a)
reasoning.                              and representations to                   Exponents VDOE
    A relationship exists between the                                                ESS (2004)
laws of exponents and scientific          Simplify monomial expressions
notation.                                  and ratios of monomial                A.2b)
expressions in which the              Mo and Po Nomial
    Operations with polynomials can be         exponents are integers, using the     VDOE ESS (2004)
represented concretely, pictorially,       laws of exponents.
and symbolically.                                                                Old Polly VDOE
   Model sums, differences, products,    ESS (2004)
    Polynomial expressions can be used         and quotients of polynomials with
to model real-world situations.            concrete objects and their related    Polynomial Puzzler
pictorial representations.
    The distributive property is the                                                 (NCTM
unifying concept for polynomial           Relate concrete and pictorial         http://illuminations.n
operations.                                manipulations that model              ctm.org/lessondetail.a
polynomial operations to their        spx?id=L798 )
    Factoring reverses polynomial              corresponding symbolic
multiplication.                                                                  Battleship for
representations.
Polynomials
    Some polynomials are prime
   Find sums and differences of          (http://quia.com )
polynomials and cannot be factored
polynomials.
over the set of real numbers.                                                    A.2c)
    Polynomial expressions can be used        Find products of polynomials. The     Functionality VDOE
to define functions and these              factors will have no more than five   ESS (2004)
functions can be represented               total terms (i.e. (4x+2)(3x+5)
Subject/Grade Level: Algebra I            Suggested Pacing for Benchmarking
Period: UNIT 5
Standard: A.2 The student will perform operations on polynomials, including
a) applying the laws of exponents to perform operations on expressions;
b) adding, subtracting, multiplying, and dividing polynomials; and
c) factoring completely first- and second-degree binomials and trinomials in one or two variables. Graphing calculators
will be used as a tool for factoring and for confirming algebraic factorizations.
Essential Knowledge                 Resources and           Suggested
Essential Understandings
and Skills                     Activities         Assessment(s)
graphically.                                   represents four terms and
(x+1)(2x2 +x+3) represents five        Promethean Board Site
 There is a relationship between the            terms).
factors of any polynomial and the x-
intercepts of the graph of its related       Find the quotient of polynomials,
function.                                      using a monomial or binomial
divisor, or a completely factored
divisor.

   Factor completely first- and
second-degree polynomials with
integral coefficients.

   Identify prime polynomials.

   Use the x-intercepts from the
graphical representation of the
polynomial to determine and
confirm its factors.
Predictor Test
Unit 6
Standard A.3
Subject/Grade Level: Algebra I               Suggested Pacing for Benchmarking
Period: UNIT 6
Standard: A.3 The student will express the square roots and cube roots of whole numbers and the square root of a monomial
algebraic expression in simplest radical form.
Essential Knowledge                 Resources and              Suggested
Essential Understandings
and Skills                     Activities            Assessment(s)
The student will use problem solving,
    A square root in simplest form is       mathematical communication,
Unit 6 Organizer        Interactive Achievement
one in which the radicand               mathematical reasoning, connections,
(argument) has no perfect square                                                Estimating Square
and representations to                  Roots VDOE ESS
factors other than one.
(2004)
   Express square roots of a whole
    A cube root in simplest form is one
number in simplest form.            Simplifying Square
in which the argument has no
perfect cube factors other than one.                                            Roots VDOE ESS
   Express the cube root of a whole    (2004)
number in simplest form.
    The cube root of a perfect cube is an
integer.
   Express the principal square root   Promethean Board Site
of a monomial algebraic
    The cube root of a nonperfect cube
expression in simplest form where
lies between two consecutive
variables are assumed to have
integers.
positive values.
    The inverse of cubing a number is
determining the cube root.

    In the real number system, the
argument of a square root must be
nonnegative while the argument of
a cube root may be any real number.
Predictor Test
Unit 6
Standard A.4c
Subject/Grade Level: Algebra I               Suggested Pacing for Benchmarking
Period: UNIT 6
Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including
a) solving literal equations (formulas) for a given variable;
b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that
are valid for the set of real numbers and its subsets;
c) solving quadratic equations algebraically and graphically;
d) solving multistep linear equations algebraically and graphically;
e) solving systems of two linear equations in two variables algebraically and graphically; and
f) solving real-world problems involving equations and systems of equations.
Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions.
Essential Knowledge                Resources and                Suggested
Essential Understandings
and Skills                      Activities             Assessment(s)
The student will use problem solving,
    A solution to an equation is the         mathematical communication,
Unit 6 Organizer         Interactive Achievement
value or set of values that can be       mathematical reasoning, connections,
substituted to make the equation                                                   Factoring for Zeros
and representations to                    VDOE ESS (2004)
true.
   Solve a literal equation (formula)
    The solution of an equation in one          for a specified variable.              Egg Launch
variable can be found by graphing                                                  (Equations of Attack
the expression on each side of the         Simplify expressions and solve         (NCTM
equation separately and finding the         equations, using the field             http://illuminations.n
x-coordinate of the point of                properties of the real numbers and     ctm.org/lessondetail.a
intersection.                               properties of equality to justify      spx?id=L738 )
simplification and solution.
    Real-world problems can be                                                         Battleship for
interpreted, represented, and solved       Solve quadratic equations.             Polynomials
using linear and quadratic                                                         (http://quia.com )
equations.                                 Identify the roots or zeros of a
    The process of solving linear and           number system as the solution(s)       Promethean Board Site
in a variety of ways, using concrete,       formed by setting the given
pictorial, and symbolic                     quadratic expression equal to zero.
representations.
   Solve multistep linear equations in
Subject/Grade Level: Algebra I              Suggested Pacing for Benchmarking
Period: UNIT 6
Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including
a) solving literal equations (formulas) for a given variable;
b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that
are valid for the set of real numbers and its subsets;
c) solving quadratic equations algebraically and graphically;
d) solving multistep linear equations algebraically and graphically;
e) solving systems of two linear equations in two variables algebraically and graphically; and
f) solving real-world problems involving equations and systems of equations.
Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions.
Essential Knowledge                Resources and                Suggested
Essential Understandings
and Skills                      Activities             Assessment(s)
 Properties of real numbers and                one variable.
properties of equality can be used to
justify equation solutions and               Confirm algebraic solutions to
expression simplification.                    linear and quadratic equations,
using a graphing calculator.

   The zeros or the x-intercepts of the       Given a system of two linear
quadratic function are the real             equations in two variables that has
root(s) or solution(s) of the               a unique solution, solve the system
quadratic equation that is formed by        by substitution or elimination to
setting the given quadratic                 find the ordered pair which
expression equal to zero.                   satisfies both equations.

   A system of linear equations with          Given a system of two linear
exactly one solution is characterized       equations in two variables that has
by the graphs of two lines whose            a unique solution, solve the system
intersection is a single point, and         graphically by identifying the
the coordinates of this point satisfy       point of intersection.
both equations.
   Determine whether a system of
   A system of two linear equations            two linear equations has one
with no solution is characterized by        solution, no solution, or infinite
the graphs of two lines that are            solutions.
parallel.
Subject/Grade Level: Algebra I              Suggested Pacing for Benchmarking
Period: UNIT 6
Standard: A.4 The student will solve multistep linear and quadratic equations in two variables, including
a) solving literal equations (formulas) for a given variable;
b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that
are valid for the set of real numbers and its subsets;
c) solving quadratic equations algebraically and graphically;
d) solving multistep linear equations algebraically and graphically;
e) solving systems of two linear equations in two variables algebraically and graphically; and
f) solving real-world problems involving equations and systems of equations.
Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions.
Essential Knowledge               Resources and                Suggested
Essential Understandings
and Skills                      Activities             Assessment(s)
 A system of two linear equations             Write a system of two linear
having infinite solutions is                  equations that models a real-world
characterized by two graphs that              situation.
coincide (the graphs will appear to
be the graph of one line), and the           Interpret and determine the
coordinates of all points on the line         reasonableness of the algebraic or
satisfy both equations.                       graphical solution of a system of
two linear equations that models a
 Systems of two linear equations can           real-world situation.
be used to model two real-world
conditions that must be satisfied            Determine if a linear equation in
simultaneously.                               one variable has one, an infinite
number, or no solutions.†

   Equations and systems of equations
can be used as mathematical models
for real-world situations.

   Set builder notation may be used to
represent solution sets of equations.                 †
Revised March 2011
Predictor Test
Unit 7
Standard A.11
Subject/Grade Level: Algebra I              Suggested Pacing for Benchmarking
Period: UNIT 7
Standard: A.11 The student will collect and analyze data, determine the equation of the curve of best fit in order to make
predictions, and solve real-world problems, using mathematical models. Mathematical models will include linear and quadratic
functions.
Essential Knowledge              Resources and                 Suggested
Essential Understandings
and Skills                     Activities               Assessment(s)
The student will use problem solving,
    The graphing calculator can be used    mathematical communication,
Unit 7 Organizer         Interactive Achievement
to determine the equation of a curve   mathematical reasoning, connections,
of best fit for a set of data.                                                     Estimating Ages of
and representations to                      Famous People
    The curve of best fit for the                                                      (mathbits.com)
   Write an equation for a curve of
relationship among a set of data
best fit, given a set of no more than   Illuminations: Line of
points can be used to make
twenty data points in a table, a        Best Fit (NCTM
predictions where appropriate.
graph, or real-world situation.         Illuminations
    Many problems can be solved by                                                     http://illuminations.n
   Make predictions about unknown          ctm.org/ActivityDetai
using a mathematical model as an
outcomes, using the equation of         l.aspx?ID=146
interpretation of a real-world
the curve of best fit.
situation. The solution must then
refer to the original real-world
   Design experiments and collect          Barbee Bungee
situation.                                 data to address specific, real-world    Jumping (NCTM
questions.
    Considerations such as sample size,                                                Illuminations
randomness, and bias should affect                                                 http://illuminations.n
   Evaluate the reasonableness of a        ctm.org/LessonDetail
experimental design.                       mathematical model of a real-           .aspx?id=L646
world situation.

Line of Best Fit
VDOE ESS (2004)
SOL TEST
Unit 8
Standard A.10
Subject/Grade Level: Algebra I               Suggested Pacing for Benchmarking
Period: UNIT 8
Standard: A.10 The student will compare and contrast multiple univariate data sets, using box-and-whisker plots.
Essential Knowledge               Resources and                Suggested
Essential Understandings
and Skills                     Activities               Assessment(s)
The student will use problem solving,
    Statistical techniques can be used to   mathematical communication,
Unit 8 Organizer        Interactive Achievement
organize, display, and compare sets     mathematical reasoning, connections,
of data.                                                                         Box-and-Whisker
and representations to                   Plots VDOE ESS
    Box-and-whisker plots can be used                                                (2004)
   Compare, contrast, and analyze
to analyze data.
data, including data from real-      Vashon-Maury Island
world situations displayed in box-   Soil Study VDOE
and-whisker plots.                   ESS (2004)

Promethean Board Site
SOL TEST
Unit 8
Standard A.9
Subject/Grade Level: Algebra I                Suggested Pacing for Benchmarking
Period: UNIT 8
Standard: A.9 The student, given a set of data, will interpret variation in real-world contexts and calculate and interpret
mean absolute deviation, standard deviation, and z-scores.
Essential Knowledge                  Resources and               Suggested
Essential Understandings
and Skills                       Activities             Assessment(s)
The student will use problem solving,
    Descriptive statistics may include                                                Unit 8 Organizer        Interactive Achievement
mathematical communication,
measures of center and dispersion.
mathematical reasoning, connections,     Good Assorted
    Variance, standard deviation, and        and representations to                   Problem Set
mean absolute deviation measure                                                   developed by Jeff
the dispersion of the data.                 Analyze descriptive statistics to    Holt of UVA (see
determine the implications for the   Instr. Specialist Sue
    The sum of the deviations of data            real-world situations from which     Jenkins)
points from the mean of a data set is        the data derive.
0.
   Given data, including data in a      Promethean Board Site
    Standard deviation is expressed in           real-world context, calculate and
the original units of measurement of         interpret the mean absolute
the data.                                    deviation of a data set.

    Standard deviation addresses the            Given data, including data in a
dispersion of data about the mean.           real-world context, calculate
variance and standard deviation of
    Standard deviation is calculated by          a data set and interpret the
taking the square root of the                standard deviation.
variance.
   Given data, including data in a
    The greater the value of the                 real-world context, calculate and
standard deviation, the further the          interpret z-scores for a data set.
data tend to be dispersed from the
mean.                                       Explain ways in which standard
    For a data distribution with outliers,       examining the formula for
the mean absolute deviation may be           standard deviation.
a better measure of dispersion than
Subject/Grade Level: Algebra I              Suggested Pacing for Benchmarking
Period: UNIT 8
Standard: A.9 The student, given a set of data, will interpret variation in real-world contexts and calculate and interpret
mean absolute deviation, standard deviation, and z-scores.
Essential Knowledge                  Resources and               Suggested
Essential Understandings
and Skills                       Activities             Assessment(s)
the standard deviation or variance.      Compare and contrast mean
absolute deviation and standard
deviation in a real-world context.
 A z-score (standard score) is a
measure of position derived from
the mean and standard deviation of
data.

   A z-score derived from a particular
data value tells how many standard
deviations that data value is above
or below the mean of the data set. It
is positive if the data value lies
above the mean and negative if the
data value lies below the mean.

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