Algebra 1 Unit Descriptions
The differences between an ex pression and an equation are further highlighted at
the beginning of Algebra I. Properties, including the distributive property,
are used to simplify expressions, and used to factor basic expressions. The
concept of absolute value is extended to studying equations in one variable with
an absolute value term. The concept of infinity is introduced, and is linked
with the number of solutions of linear equations in one and two variables.
Graphs
of linear equations are used to develop the concept of systems of linear
equations. Methods of solving systems of linear equations are explored. The
concepts of function and exponent are expanded to the study of quadratic
functions. Quadratic functions are analyzed and quadratic equations are solved.
Connections are made between the xintercepts of a quadratic function and the
solutions to a quadratic equation. The concepts of expression and exponent
are further expanded to include polynomials. Polynomials are added, subtracted ,
and multiplied. Methods of factoring polynomials, especially quadratics,
are discussed in depth, and the need for factoring is highlighted. The concept
of the rational numbers is used to develop the concept of rational expressions
with polynomials in the numerator and denominator . Operations on rational
expressions are performed. The concept of inequalities is then introduced as a
way to solidify understanding and link many previous concepts. Mathematical
reasoning is weaved throughout the course both in justifying the steps to
simplify expressions, solve equations and inequalities and in solving applied
problems requiring solving for unknowns.
Unit Organization:
Unit 1: Using Properties to Simplify and Solve
Unit 2: Linear Functions
Unit 3: Systems of Linear Equations
Unit 4: Introduction to Quadratic Equations
Unit 5: Solving Quadratics
Unit 6 Polynomials and The Role of the Distributive Property
Unit 7: Rational Expressions
Unit 8: Working with Inequalities
Unit 10: CST Review
Textbook: Prentice Hall: Algebra 1 (California Edition 2001)
Unit 1: Using Properties to Simplify and Solve
This unit lays the foundation for proving geometric relationships by developing
the effective thinking needed to construct mathematically accurate logical
arguments and proofs. Inductive reasoning is introduced as the type of thinking
needed to analyze patterns and create conjectures. Counterexamples are
used to prove some conjectures false. Different forms of logical statements are
explored, including the converse, inverse, contra positive , and biconditional
statements. Counterexamples are used again to show that a logical statement is
false. Logical statements are then used in defining basic geometric objects
(i.e. points, lines, and planes), postulates (i.e. addition, etc), and
properties (i.e. reflexive, symmetric, transitive). The understanding of
segments is explored
with constructions (i.e. copying of segments) and deductive reasoning is used to
prove basic geometric statements, including those about segments. Two
column proofs are introduced and a connection is made between this type of
proof, and written justification of steps used in an algebra problem. Multiple
types of proofs are used to prove statements about segment congruence, including
twocolumn proofs and paragraph proofs. As an introduction to
coordinate geometry, the distance formula and the midpoint are derived and
organized as proofs. The definition of equidistant is given and illustrated in
the
coordinate plane using the distance formula.
# 
Learning Targets 
Stand
ard 
Textbook 
Active Practice 
1A 
Evaluate expressions using the order
of operations correctly and justify each step 
1.0 
1.1: pg 4 – 8
# 37  50 

1B 
Use variables and operations to
construct expressions that represent mathematical
situations and explain your reasoning 
5.0 
1.6: pg 30 #1 – 51
3.4: pg 130 – 135 #1
 18 

1C 
Simplify an expression by combining
like terms and explain how the expressions are
affected by the algebraic properties. 
4.0 
1.2: pg. 914
1.4: pg 1922
5.7: pg 233
#5  24 

1D 
Use the multiplication property of
exponents while using the distributive property to simplify
expressions. 
2.0 
5.1: pg 204 – 207
5.9 ( not binomials ):
pg 242 #1 – 20 

1E 
Explain how the process of factoring
out the GCF of an expression is related to the
distributive property. 
4.0 
6.1: pg 262 – 264
#22  40 

1F 
Explain how the concept of
equivalence is used in simplifying algebraic expressions. 
4.0 
2.10: pg 102 – 105
# 1 – 14 

1G 
Differentiate between and explain the
process of solving single and twostep equations (in
one variable). 
3.0,
5.0 
3.1 – 3.3
pg 127 #1  14 

1H 
Simplify equations before solving
multistep equations 
4.0 
3.3
pg 128 #32  41 

1I 
Solve equations with variables on
both sides of the equation and explain how the algebraic
properties are used in the process 
4.0,
1.0 
3.5 pg 136 – 138
#1  28 

1J 
Explain the meaning of an absolute
value and solve multistep equations containing
absolute values. 
3.0 
9.3
pg 412 #22  33 

1K 
Use the algebraic properties to
justify solving multistep single variable equations. 
1.1 
3.3
Pg 128 #43  56 

1K 
Describe the additive and
multiplicative identity and inverse properties, and use them to
justify solving multistep equations 
1.1 
1.2 pg 10
Pg 128 #43  56 

1L 
Use relationships within word
problems to solve complex applied problems 
5.0 
Station practice
worksheet
3.4: pg 134 #13  29 

Essential Standards:
Algebra 1:
1.0: Students identify and use the arithmetic properties of subsets of integers
and rational, irrational, and real numbers , including closure properties for the
four basic arithmetic operations where applicable
3.0: Students solve equations and inequalities involving absolute values
4.0: Students simplify expressions before solving linear equations and
inequalities in one variable, such as 3(2x5) + 4(x2) = 12
5.0: Students solve multistep problems, including word problems, involving
linear equations and linear inequalities in one variable and provide
justification
for each step.
Supporting Standards:
Algebra 1:
1.1: Students use properties of numbers to demonstrate whether assertions are
true or false.
2.0: Students understand and use such operations as taking the opposite, finding
the reciprocal, taking a root and raising to a fractional power. They
understand and use the rules of exponents
11.0: Students apply basic factoring techniques to second and simple
thirddegree polynomials. These techniques include finding a common factor for
all
terms in a polynomial , recognizing the difference or two squares, and
recognizing perfect squares of binomials.
15.0: Students apply algebraic techniques to solve rate problems, work problems,
and percent mixture problems
24.0: Students use and know simple aspects of a logical argument
25.0: Students use properties of the number system to judge the validity of
results, to justify each step of a procedure, and to prove or disprove
statements.
Unit 2: Linear Functions
Up to this point, only equations in one variable have been studied. In this
unit, the concept of a linear equation in two variables is introduced. The
meaning
of an equation in two variables is the same as in one variable. It is still a
statement that two expressions either of which now may have two variables, are
equal. Furthermore, whereas a solution to an equation in one variable is a
single number that makes the equation true, a solution to an equation in two
variables is an ordered pair that makes the equation true. This means that the
solution set of an equation in two variables is the set of ordered pairs that
makes the equation true. The graph of an equation is simply the graph of every
ordered pair in the solution set. Equations in two variables can also be
manipulated using the properties of equality, just like with equations in one
variable. These properties are used to solve an equation in two variables for
one
of the variables in terms of the other.
A relation is then defined as a set of ordered pairs. The solution set of an
equation in two variables is introduced as an example of a relation. A relation
is
also presented as a rule that given an input value produces one or more output
values based on that rule. For every input to a relation, there may be zero,
one, or more than one output. A variety of different relations are discussed
based on a given rule (e.g. add 2 to the input, or take the square root of the
input)
and different representations of relations are discussed including sets of
ordered pairs, mappings, tables, equations, and graphs.
There is a special type of relation that is studied throughout all areas of
mathematics, namely a function. A function is defined as a relation such that
for
every input, there exists one and only one output. The definition of a function
is then used to determine if previous examples of relations are functions or
not.
Since a function is just a special type of relation, the various representations
used with relations (i.e. mapping, table, etc.) are also used with functions.
The
independent variable is defined as the input variable, and the dependent
variable is defined as the output variable. The dependent variable literally
depends
upon the independent variable. Using this notion of independence, tables are
created for given functions (written as equations) using any numbers for the
inputs. The graphs of these functions are then drawn using these tables.
Throughout this course and future courses, functions are classified by certain
properties. This unit focuses on developing properties of linear functions. A
linear function is a function that is defined by an equation of the form y = ax
+ b, where x is the independent variable, and y is the dependent variable.
Functions tables are then used to graph a wide variety of linear functions. It
is emphasized that the graph of every linear function is a line. This naturally
leads to a discussion of how to classify and describe these lines using their
“steepness”. This notion is formally introduced by defining the slope a line
joining
two points as the ratio of the change in y and the change in x. The formula for
the slope, m, of the line joining the points
and is developed,
namely , and used to find the slope of the
graphs of a wide variety of linear functions, including the slopes of vertical
and horizontal lines.
It is observed that the slope of the graph of a linear function appears as the
coefficient of x when the function is written as y = ax + b. Furthermore, the
yvalue of the point where the line intersects the yaxis is identified as the
constant term. This point is called the yintercept. From here, the
slopeintercept
form of the equation of the line is defined as y = mx + b, where m is the slope,
and b is the yintercept. The graphs of linear functions in slopeintercept
form are drawn. The pointslope form of a line,
, is introduced and used to write equations of lines when given a point and a
slope, or
when given two points. Students see that no matter the form that is chosen,
information can be substituted into either the pointslope form or the slope
intercept form to write an equation for a line.
In the development of the slopeintercept form, the
yintercept of a graph was defined. The xintercept is similarly defined as the
point where the graph
crosses the xaxis. The intercepts of a linear function are found by two
methods: first, by identifying intercepts from the graph of a line, and second
by
recognizing that one of the coordinates is 0 at each intercept and therefore can
be used to find the intercepts algebraically (e.g. find the xintercept by
plugging in 0 for y and solving for x).
The roots of a function are then defined as the input values that make the
function have a value of 0. This is represented graphically by the xintercepts
of
the graph of a function. A method for solving for the roots of a linear function
is illustrated and connected to methods of solving linear equations in one
variable.
The students investigate the slopes of parallel and perpendicular lines before
they begin the next unit on systems of equations.
# 
Learning Targets 
Stand
ard 
Textbook 
Active Practice 
2A 
Solve an equation for one variable in
terms of another. 
6.0 
Pg 326
# 1  18 

2B 
Explain the difference between an
equation with one variable and those with two variables. 
17.0 
notes 

2C 
Find the domain and range of a set of
data when given the ordered pairs 
17.0 
12.1 pg 538
#1  6 

2D 
Graph a line by building a function
table and use the function table to describe how the line
represents a function. 
17.0, 18.0 
7.3 pg 316
# 1  15 

2E 
Determine graphically whether a point
is a solution to a linear equation. 
7.0 
Envelope activity 

2F 
Determine algebraically whether a
point lies on the line
(whether or not the ordered pair is a solution to the equation). 
7.0 
7.2 pg 311
#1  15 

2G 
Derive the formula for slope and use
it to find the slope of a given line. 
6.0 
7.4 pg 321
# 11  20 

2H 
Graph a line using slope intercept
form and explain the process of transforming the
equation into a graph. 
6.0 
7.5 pg 326
# 28  39 

2I 
Find the x and y intercept of a
linear equation and explain the difference between the two. 
6.0 
7.3 pg 316
#16  33 

2J 
Write the equation for a line when
given the slope and a point on the line and explain
which form (slopeintercept/pointslope) I prefer and why. 
7.0 
7.6 pg 331
# 1  12 

2K 
Write the equation for a line when
given two points that lie on the line and explain the
process one goes through to do this. 
7.0 
7.6 pg 331
# 13  22 

2L 
Define the root/zero/solution of a
linear function and find the roots/zeros/solutions of a
linear function. 
6.0 
Roots worksheet 

Essential Standards:
6.0: Students graph a linear equation and compute the x and y intercepts
(e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by
linear
inequality (e.g., they sketch the region defined by 2x + 6y < 4).
7.0: Students verify that a point lies on a line, given an equation on the line.
Students are able to derive linear equations by using the pointslope formula.
17.0: Students determine the domain of independent variables and the range of
dependent variables defined by a graph, a set of ordered pairs, or a symbolic
expression
Supporting Standards:
Algebra 1:
8.0: Students understand the concepts of parallel lines and perpendicular lines
and how those slopes are related. Students are able to find the equation of a
line perpendicular to a given line that passes through a given point.
16.0: Students understand the concepts of a relation and a function, determine
whether a given relation defines a function, and give pertinent information
about given relations and functions.
18.0: Students determine whether a relation defined by a graph, a set of ordered
pairs, or symbolic expression is a function and justify the conclusion
21.0: Students graph quadratic functions and know that their roots are
xintercepts
24.0: Students use and know simple aspects of a logical argument.
25.0: Students use properties of the number system to judge the validity of
results, to justify each step of a procedure, and to prove or disprove
statements.