Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations, that use positive and negative integers and combinations of these operations.
Write and solve one-step linear equations in one variable.
Write and evaluate an algebraic expression for a given situation, using up to three variables.
Apply algebraic order of operations and the commutative, associative, and distributive properties to evaluate expressions; and justify each step in the process.
Solve problems manually by using the correct order of operations or by using a scientific calculator.
Use variables in expressions describing geometric quantities (e.g., P = 2w + 2l, A = 1/2 bh, C = pi d - the formulas for the perimeter of a rectangle, the area of a triangle, and the circumference of a circle, respectively).
Express in symbolic form simple relationships arising from geometry.
Use the properties of complementary and supplementary angles and the sum of the angles of a triangle to solve problems involving an unknown angle.
Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g., three less than a number, half as large as area A).
Use the correct order of operations to evaluate algebraic expressions such as 3(2x + 5)².
Simplify numerical expressions by applying properties of rational numbers (e.g., identity, inverse, distributive, associative, commutative) and justify the process used.
Use algebraic terminology (e.g., variable, equation, term, coefficient, inequality, expression, constant) correctly.
Represent quantitative relationships graphically and interpret the meaning of a specific part of a graph in the situation represented by the graph.
Graph functions of the form y = nx² and y = nx³ and use in solving problems.
Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio ("rise over run") is called the slope of a graph.
Plot the values of quantities whose ratios are always the same (e.g., cost to the number of an item, feet to inches, circumference to diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the quantities.
Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results.
Solve multistep problems involving rate, average speed, distance, and time or a direct variation.
Students use properties of numbers to demonstrate whether assertions are true or false.
Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x - 5) + 4(x - 2) = 12.
Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.
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).
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.
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.
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.
Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques.
associates verbal names, written word names, and standard numerals with integers, fractions, decimals; numbers expressed as percents; numbers with exponents; numbers in scientific notation; radicals; absolute value; and ratios.
understands and explains the effects of addition, subtraction, multiplication, and division on whole numbers, fractions, including mixed numbers, and decimals, including the inverse relationships of positive and negative numbers.
represents and applies geometric properties and relationships to solve real-world and mathematical problems.
identifies and plots ordered pairs in all four quadrants of a rectangular coordinate system (graph) and applies simple properties of lines.
describes a wide variety of patterns, relationships, and functions through models, such as manipulatives, tables, graphs, expressions, equations, and inequalities.
creates and interprets tables, graphs, equations, and verbal descriptions to explain cause-and-effect relationships.
represents and solves real-world problems graphically, with algebraic expressions, equations, and inequalities.
uses algebraic problem-solving strategies to solve real-world problems involving linear equations and inequalities.
compares experimental results with mathematical expectations of probabilities.
determines odds for and odds against a given situation.
associates verbal names, written word names, and standard numerals with integers, rational numbers, irrational numbers, real numbers, and complex numbers.
understands the relative size of integers, rational numbers, irrational numbers, and real numbers.
understands that numbers can be represented in a variety of equivalent forms, including integers, fractions, decimals, percents, scientific notation, exponents, radicals, absolute value, and logarithms.
understands and explains the effects of addition, subtraction, multiplication, and division on real numbers, including square roots, exponents, and appropriate inverse relationships.
selects and justifies alternative strategies, such as using properties of numbers, including inverse, identity, distributive, associative, transitive, that allow operational shortcuts for computational procedures in real-world or mathematical problems.
represents and applies geometric properties and relationships to solve real-world and mathematical problems including ratio, proportion, and properties of right triangle trigonometry.
using a rectangular coordinate system (graph), applies and algebraically verifies properties of two and three-dimensional figures, including distance, midpoint, slope, parallelism, and perpendicularity.
describes, analyzes, and generalizes relationships, patterns, and functions using words, symbols, variables, tables, and graphs.
determines the impact when changing parameters of given functions.
represents real-world problem situations using finite graphs, matrices, sequences, series, and recursive relations.
uses systems of equations and inequalities to solve real-world problems graphically, algebraically, and with matrices.
determines probabilities using counting procedures, tables, tree diagrams, and formulas for permutations and combinations.
determines the probability for simple and compound events as well as independent and dependent events.
Write and evaluate mathematical expressions that correspond to given situations.
Write, solve, and graph one- and two- step linear equations and inequalities.
Works backward with two-step function rules to undo expressions.
Solve problems given a formula.
Apply the Commutative, Associative, and Distributive Properties to show that two expressions are equivalent.
Construct and analyze tables, graphs and equations to describe linear functions and other simple relations using both common language and algebraic notation.
Find the perimeters and areas of composite two-dimensional figures, including non-rectangular figures (such as semicircles) using various strategies.
Determine a missing dimension of a plane figure or prism, given its area or volume and some of the dimensions, or determine the area or volume given the dimensions.
Solve problems involving similar figures.
Graph proportional relationships and identify the unit rate as the slope of the related linear function.
Use and justify the rules for adding, subtracting, multiplying, dividing, and finding the absolute value of integers.
Add, subtract, multiply, and divide integers, fractions, and terminating decimals, and perform exponential operations with rational bases and whole number exponents including solving problems in everyday contexts.
Formulate and use different strategies to solve one-step and two-step linear equations, including equations with rational coefficients.
Use the properties of equality to represent an equation in a different way and to show that two equations are equivalent in a given context.
Determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and apply these relationships to solve problems.
Identify and plot ordered pairs in all four quadrants of the coordinate plane.
Create and interpret tables, graphs, and models to represent, analyze, and solve problems related to linear equations, including analysis of domain, range and the difference between discrete and continuous data.
Interpret the slope and the x- and y-intercepts when graphing a linear equation for a real-world problem.
Use tables, graphs, and models to represent, analyze, and solve real-world problems related to systems of linear equations.
Identify the solution to a system of linear equations using graphs.
Translate among verbal, tabular, graphical and algebraic representations of linear functions.
Compare the graphs of linear and non-linear functions for real-world situations.
Use similar triangles to solve problems that include height and distances.
Classify and determine the measure of angles, including angles created when parallel lines are cut by transversals.
Demonstrate that the sum of the angles in a triangle is 180-degrees and apply this fact to find unknown measure of angles, and the sum of angles in polygons.
Validate and apply Pythagorean Theorem to find distances in real world situations or between points in the coordinate plane.
Solve literal equations for a specified variable.
Perform operations on real numbers (including integer exponents, radicals, percents, scientific notation, absolute value, rational numbers, and irrational numbers) using multi-step and real world problems.
Solve linear equations in one variable that include simplifying algebraic expressions.
Identify and apply the distributive, associative, and commutative properties of real numbers and the properties of equality.
Solve literal equations for a specified variable.
Solve and graph simple and compound inequalities in one variable and be able to justify each step in a solution.
Symbolically represent and solve multi-step and real-world applications that involve linear equations and inequalities.
Solve and graph the solutions of absolute value equations and inequalities with one variable.
Rewrite equations of a line into slope-intercept form and standard form.
Graph a line given any of the following information: a table of values, the x- and y-intercepts, two points, the slope and a point, the equation of the line in slope-intercept form, standard form, or point-slope form.
Determine the slope, x-intercept, and y-intercept of a line given its graph, its equation, or two points on the line.
Write an equation of a line given any of the following information: two points on the line, its slope and one point on the line, or its graph. Also, find an equation of a new line parallel to a given line, or perpendicular to a given line, through a given point on the new line.
Write an equation of a line that models a data set and use the equation or the graph to make predictions. Describe the slope of the line in terms of the data, recognizing that the slope is the rate of change.
Graph a linear equation or inequality in two variables with and without graphing technology. Write an equation or inequality represented by a given graph.
Use a graph to approximate the solution of a system of linear equations or inequalities in two variables with and without technology.
Solve systems of linear equations and inequalities in two and three variables using graphical, substitution, and elimination methods.
Solve real-world problems involving systems of linear equations and inequalities in two and three variables.
Simplify algebraic ratios.
Add, subtract, multiply, and divide rational expressions.
Simplify complex fractions.
Solve algebraic proportions.
Solve rational equations.
Identify removable and non-removable discontinuities and vertical, horizontal, and oblique asymptotes of a graph of a rational function, find the zeros, and graph the function.
Solve real-world problems involving rational equations (mixture, distance, work, interest, and ratio).