# Algebra 1 Unit Descriptions &amp; Learning Targets

Unit 6: Polynomials and The Role of the Distributive Property
A quadratic expression is just a special case of a polynomial expression. A polynomial is then defined, and examples are used to illustrate the connections
between quadratics and polynomials . Important vocabulary for polynomials is defined, including degree and leading coefficient. Operations on polynomials
are mastered, including addition, subtraction, and multiplication. Multiplication of polynomials is not strictly held to binomials and the FOIL method, rather the
concept of distribution is used to show that polynomials of more than two terms can also be multiplied.

The focus of the unit then shifts to a discussion of factoring. As an introduction to factoring, the case of the difference of squares is discussed in terms of
the solution to the equation . Namely, that the since the solutions to that equation, a and –a, are just the roots of the function, the equation can
be written as (x + a)(x - a) = 0. Factoring out the GCF is then reviewed from the first unit and expanded to include more complex polynomials. This
concept is expanded to develop the method of factoring by grouping. For example, the expression x3 + x2 + 2x + 2 is written as x2(x +1) + 2(x +1) and
then the (x +1) term is factored out so that the final factorization is (x2 + 2)(x +1).

This method is then applied to factor quadratic expressions of the form by first multiplying a- c and then determining the pair of factors of
ac whose sum is b. Then, b is rewritten using this pair of factors, and the method of factoring by grouping is used. For example, the expression
2x2 + 5x + 2 is written as 2x2 + 4x + x + 2, and then factored into (2x +1)(x + 2). Factoring quadratics of the form is introduced only as a
special case of the standard form of a quadratic.

The use of the zero product property is employed throughout the section on factoring in order to solve factorable quadratic equations. An emphasis is
placed on the distinction between factoring an expression and solving an equation. Finally, division of polynomials is explored and the role of factoring in
reducing such an expression to lowest terms is discussed.

 # Learning Targets Stand ard Textbook Active Practice 6A Define polynomial, identify the degree of polynomials, and provide examples of expressions that are polynomials. 10.0 6B Add and subtract polynomials and explain how the two operations on polynomials are related. 10.0 6C Multiply two or more binomials and explain the role of the distributive property 10.0 6D Multiply many polynomials and explain how the distributive property is related to polynomial multiplication. 10.0 6E Explain the relationship between the multiplication of polynomials and factoring polynomials. 10.0, 11.0 6F Factor a difference of two squares polynomial . 11.0 6G Factor out the GCF from polynomials with many terms. 11.0 6H Factor a polynomial expression by grouping and identify algebraic properties that are used in the process. 11.0 6I Use factor by grouping to factor quadratic expressions that are in standard form (ax2+bx +c). 11.0 6J Solve quadratic equations by factoring and using the Zero Product Property and explain the process of doing so. 11.0 14.0 6K Use factoring to find the roots of a factorable quadratic equation. 11.0 6L Explain the difference between solving a quadratic equation by factoring and factoring a quadratic expression. 11.0, 14.0 6M Use factoring to simplify expressions (reducing) involving polynomial division. 10, 11.0 6N Explain and provide justification for my reasoning why polynomials cannot be divided “term by term.” 10.0

Essential Standards (CA):
10.0: Students add, subtract, multiply, and divide monomials and polynomials. Students solve multi-step problems, including word problems, by using these
techniques.
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.

Supporting Standards: Algebra 1:
12.0: Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.
21.0: Students graph quadratic functions and know that their roots are the x-intercepts.
14.0: Students solve a quadratic equation by factoring or completing the square.
22.0: Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in
zero, one, or two points.
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 7: Rational Expressions
Rational expressions are built by dividing two polynomial expressions, called
N(x) for numerator and
D(x) for denominator. The concept of writing a
rational expression in lowest terms using factoring N(x) and D(x) is expanded from the previous unit, and provides a foundation for performing the operations
of multiplication and division on rational expressions. A comparison of different methods of multiplying rational expressions (i.e. multiply across, or factor
then simplify) illustrates another need for factoring polynomials. The operations of multiplication and division on rational expressions are consistently
compared with the same operations on rational numbers.

Continuing with the four operations, the addition and subtraction of rational expressions is discussed for like denominators . Emphasis is placed on
distributing the negative sign in the numerator of the second rational expression when subtracting. The concept of least common denominator (LCD) is
reviewed with rational numbers, and the LCD of multiple rational expressions is found, and used to rewrite expressions with common denominators. Addition
and subtraction of rational expressions with unlike denominators is discussed with increasingly more complex examples, while still bridging connections to
addition and subtraction of rational numbers.

 # Learning Targets Stand ard Textbook Active Practice 7A Explain the process of simplifying a rational expression by factoring and reducing to lowest terms. 12.0 7B Explain why factoring is needed in multiplication and division, as opposed to just multiplying “straight across.” 12.0 7C Multiply rational expressions and simplify to lowest terms. 13.0 7D Divide rational expressions and can explain the process of doing so. 13.0 7E Explain the relationship between the multiplication of and division of rational expressions. 13.0 7F Add and subtract rational expressions with like denominators. 13.0 7G Find the least common denominator of two or more rational expressions and explain the process of how this is done. 12.0 7H Use the least common denominator to write two or more rational expressions with the same denominator and then add or subtract them. 12.0, 13.0 7I Explain how to simplify rational expressions that involve addition, subtraction, multiplication, and division 13.0

Essential Standards (CA):
12.0: Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.
13.0: Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging
problems by using these techniques.

Supporting Standards:
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 8: Working with Inequalities
This unit is the capstone of the course, in which many essential concepts are reviewed and expanded in the context of inequalities. The unit begins with
situational word problems and the writing of inequalities that represent those particular situations. The graphs of basic inequalities on a number line are
reviewed. This knowledge is then used to solve and graph the solution set of a linear inequality in one variable. It is emphasized that whenever multiplying
or dividing by a negative number in any inequality, the direction of the inequality sign changes. Problems are solved using linear inequalities in one variable.
The concept of absolute value as distance from 0 is revisited. Compound inequalities are used to solve absolute value inequalities in one variable.
Knowledge of graphing linear equations provides the foundation for graphing solution sets of inequalities in two variables. The solution set of a linear
inequality in one variable is graphed in the coordinate plane . The graphs of inequalities in one and two variables are compared (e.g. compare
and ). A comparison of the behavior of the boundary point and boundary line for both one and two variable linear inequalities is explored. The
solution sets of systems of linear inequalities are also identified and graphed. These systems in two variables may have more than inequalities. Continuing
the discussion of inequalities in two variables, quadratic inequalities are explored, as well as systems mixed with quadratic and linear inequalities.

 # Learning Targets Stand ard Textbook Active Practice 8A Use inequalities in one and two variables to algebraically represent real world situations and explain reasoning behind the mathematical model . 5.0 8B Draw the graph of an inequality based on a verbal description and justify the diagram 4.0 8C Determine the solution set of a linear inequality in one variable and graph 4.0 8D Solve absolute value inequalities in one variable and explain how they are visually represented on the number line and why absolute value is necessary to describe the situation. 3.0 8E Graph the solution set of a linear inequality and explain the process used to find the solution 6.0 8F Explain the relationship between the boundary point of a one variable inequality and the boundary line of a two variable inequality 6.0 8G Graph the solution set of a system of linear inequalities, including systems with two variables, and two inequalities. 9.0 8H Graph the solution set of a quadratic inequality and justify that the solution set is accurate 21.0 8I Graph systems of inequalities including systems with quadratic inequalities and relate the concept of the boundary curve of a quadratic inequality, with the concepts of the boundary point and boundary line. 21.0 8J Solve real world problems that require mathematical models involving inequalities 5.0

Essential Standards (CA):
3.0: Students solve equations and inequalities involving absolute values.
4.0: Students simplify expressions prior to solving linear equations and inequalities in one variable, such as 3(2x – 5) + 4(x – 2) = 12.
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).
9.0: Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve
a system of two linear inequalities in two variables and to sketch the solution sets.

Supporting Standards:
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.
15.0: Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.
21.0: Students graph quadratic functions and know that their roots are the 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.

Unit 10: CST Review Unit
In this final unit, the connections that have been formed from unit to unit are solidified. The relationships between all the skill sets and concepts that have
been learned throughout the course are communicated. As a way of differentiating instruction, understanding of skills and concepts from the course is self-assessed,
and an individualized plan is developed to address any deficiencies.

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