Algebra 1 Unit Descriptions & Learning Targets

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 x-intercepts 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 two-column 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.

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(2x-5) + 4(x-2) = 12
5.0: Students solve multi-step 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 third-degree 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
y-value of the point where the line intersects the y-axis is identified as the constant term. This point is called the y-intercept. From here, the slope-intercept
form of the equation of the line is defined as y = mx + b, where m is the slope, and b is the y-intercept. The graphs of linear functions in slope-intercept
form are drawn. The point-slope 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 point-slope form or the slope
intercept form to write an equation for a line.

In the development of the slope-intercept form, the y-intercept of a graph was defined. The x-intercept is similarly defined as the point where the graph
crosses the x-axis. 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 x-intercept 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 x-intercepts 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.

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 point-slope 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 x-intercepts
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.