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Testing

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Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

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Construct a function to model a linear relationship between two quantities. Determine the rate of change  and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

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Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.

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Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

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Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

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Use functions to model relationships between quantities.

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Use functions to model relationships between quantities.

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Define, evaluate, and compare functions.

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Define, evaluate, and compare functions.

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Solve real-world and mathematical problems leading to two linear equations in two variables.

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Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.

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Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

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Analyze and solve pairs of simultaneous linear equations.

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Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

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Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

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Solve linear equations in one variable.

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Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

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Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

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Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

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Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.

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Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

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Know and apply the properties of integer exponents to generate equivalent numerical expressions.

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Analyze and solve linear equations and pairs of simultaneous linear equations.

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Analyze and solve linear equations and pairs of simultaneous linear equations.

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Understand the connections between proportional relationships, lines, and linear equations.

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Understand the connections between proportional relationships, lines, and linear equations.

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Work with radicals and integer exponents.

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Work with radicals and integer exponents.

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Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.

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Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.

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Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

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Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.

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Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.

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Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

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Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

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Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

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Use properties of operations to generate equivalent expressions.

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Use properties of operations to generate equivalent expressions.

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Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.

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Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

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Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

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Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

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Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

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Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).

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Apply the properties of operations to generate equivalent expressions.

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Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).

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Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.

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Write expressions that record operations with numbers and with letters standing for numbers.

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Write, read, and evaluate expressions in which letters stand for numbers.

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Write and evaluate numerical expressions involving whole-number exponents.

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Represent and analyze quantitative relationships between dependent and independent variables.

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Represent and analyze quantitative relationships between dependent and independent variables.

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Reason about and solve one-variable equations and inequalities.

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Reason about and solve one-variable equations and inequalities.

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Apply and extend previous understandings of arithmetic to algebraic expressions.

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Apply and extend previous understandings of arithmetic to algebraic expressions.