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AI.1
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:
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AI.1.1
Students use properties of numbers to demonstrate whether assertions are true or false.
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AI.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.
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AI.3.0
Students solve equations and inequalities involving absolute values.
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AI.4.0
Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x - 5) + 4(x - 2) = 12.
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AI.5.0
Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.
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AI.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).
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AI.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.
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AI.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.
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AI.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.
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AI.10.0
Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques.
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AI.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.
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AI.12.0
Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.
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AI.13.0
Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques.
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AI.14.0
Students solve a quadratic equation by factoring or completing the square.
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AI.15.0
Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.
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AI.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.
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AI.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.
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AI.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.
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AI.19.0
Students know the quadratic formula and are familiar with its proof by completing the square.
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AI.20.0
Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations.
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AI.21.0
Students graph quadratic functions and know that their roots are the x-intercepts.
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AI.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.
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AI.23.0
Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.
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AI.24.0
Students use and know simple aspects of a logical argument:
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AI.24.1
Students explain the difference between inductive and deductive reasoning and identify and provide examples of each.
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AI.24.2
Students identify the hypothesis and conclusion in logical deduction.
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AI.24.3
Students use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion.
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AI.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:
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AI.25.1
Students use properties of numbers to construct simple, valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions.
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AI.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.
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AI.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.
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