Algebra I
Symbolic reasoning and calculations with symbols are central in
algebra. Through the study of algebra, a student develops an understanding of
the symbolic language of mathematics and the sciences. In addition, algebraic
skills and concepts are developed and used in a wide variety of
problem-solving situations.
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:
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.
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 multistep problems, including word problems, involving
linear equations and linear inequalities in one variable and provide
justification for each step.
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 of the
line. Students are able to derive linear equations by using the point-slope
formula.
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.
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.
10.0
Students add, subtract, multiply, and divide monomials and polynomials.
Students solve multistep 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 of two squares, and
recognizing perfect squares of binomials.
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.
14.0
Students solve a quadratic equation by factoring or completing the
square.
15.0
Students apply algebraic techniques to solve rate problems, work
problems, and percent mixture problems.
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.
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.
18.0
Students determine whether a relation defined by a graph, a set of
ordered pairs, or a symbolic expression is a function and justify the
conclusion.
19.0
Students know the quadratic formula and are familiar with its proof by
completing the square.
20.0
Students use the quadratic formula to find the roots of a second-degree
polynomial and to solve quadratic equations.
21.0
Students graph quadratic functions and know that their roots are the
x-intercepts.
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.
23.0
Students apply quadratic equations to physical problems, such as the motion
of an object under the force of gravity.
24.0
Students
use and know simple aspects of a logical argument:
24.1
Students explain the difference between inductive and deductive reasoning
and identify and provide examples of each.
24.2
Students identify the hypothesis and conclusion in logical deduction.
24.3
Students use counterexamples to show that an assertion is false and
recognize that a single counterexample is sufficient to refute an assertion.
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:
25.1
Students use properties of numbers to construct simple, valid arguments
(direct and indirect) for, or formulate counterexamples to, claimed assertions.
25.2
Students judge the validity of an argument according to whether the
properties of the real number system and the order of operations have been
applied correctly at each step.
25.3
Given a specific algebraic statement involving linear, quadratic, or
absolute value expressions or equations or inequalities, students determine
whether the statement is true sometimes, always, or never.