Teaching and Learning

Pre-Requisite:  7th Grade Math Extensions

• Understand the connections between proportional relationships, lines, and linear equations.
• Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.
• 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.

• Create equations that describe numbers or relationships
• Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
• Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
• Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.
• Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.

• Understand solving equations as a process of reasoning and explain the reasoning
• Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

• Solve equations and inequalities in one variable
• Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
• Solve quadratic equations in one variable.
• Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions. Derive the quadratic formula from this form.
• Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation.

• Analyze and solve linear equations and pairs of simultaneous linear equations.
• 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).
• Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
• Analyze and solve pairs of simultaneous linear equations.
• 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.
• Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.
• Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
• Solve real-world and mathematical problems leading to two linear equations in two variables.

• Represent and solve equations and inequalities graphically
• Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
• Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear and polynomial.
• Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

• Interpret the structure of expressions
• Interpret expressions that represent a quantity in terms of its context.
• Interpret parts of an expression, such as terms, factors, and coefficients.
• Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.

• Write expressions in equivalent forms to solve problems
• Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
• Factor a quadratic expression to reveal the zeros of the function it defines.
• Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

• Perform arithmetic operations on polynomials
• Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Quantities

• Reason quantitatively and use units to solve problems.
• Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

Functions

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

• Use functions to model relationships between quantities.
• 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.
• 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.

• Understand the concept of a function and use function notation
• Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
• Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
• Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

• Interpret functions that arise in applications in terms of the context
• Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
• Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

• Construct and compare linear, quadratic, and exponential models and solve problems
• Distinguish between situations that can be modeled with linear functions and with exponential functions.
• Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
• Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
• Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
• Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Expressing Geometric Properties with Equations

• Use coordinates to prove simple geometric theorems algebraically
• Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

Statistics and Probability

• Investigate patterns of association in bivariate data.
• Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
• Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
• Fit a function to the data; use functions fitted to data to solve problems in the context of the data.
• Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.

• Interpret linear models
• Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
• Compute (using technology) and interpret the correlation coefficient of a linear fit.
• Distinguish between correlation and causation.