Grades 9, 10, 11, 12

Algebra

Statistics and Probability

Functions

Numbers and Quantity

interpret expressions that represent a quantity in terms of its context

interpret parts of an expression such as terms, factors, and coefficients, in context

interpret the meaning of given formulas or expressions in context of individual terms or factors when given in situations which utilize the formulas or expressions with multiple terms and/or factors

create equations and inequalities in one variable and use them to solve problems; Include equations arising from linear, quadratic, and exponential functions - integer inputs only

create linear or exponential 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 data points as possible (a solution) or not possible (non-solution) under the established constraints

rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations (e.g., rearrange Ohm's law V=IR to highlight resistance R; rearrange the formula for the area of a circle to highlight the radius)

justify the steps of a simple one-solution equation using algebraic properties and the properties of real numbers. Justify each step, or if given two or more steps of an equation, explain the progression from one step to the next using properties.

solve linear equations and inequalities in one variable, including equations with coefficients represented by letters (Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple exponential equations that rely only on application of the laws of exponents

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 (elimination method)

solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables

demonstrate that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane

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 exponential functions

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

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 ax² + bx + c = 0

solve quadratic equations by inspection, taking square roots, factoring, completing the square, and the quadratic formula, as appropriate to the initial form of the equation (limit to real number solutions)

use the structure of an expression to rewrite it in different equivalent forms.

choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression

factor any quadratic expression to reveal the zeroes of the function it defines

complete the square in a quadratic expression to reveal the maximum and minimum value of the function defined by the expression

add, subtract, and multiply polynomials; understand that polynomials form a system analogous to the integers in that they are closed under these operations

represent data with plots on the real number line (dot plots, histograms, and box plots)

use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range) of two or more different data sets

interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers)

summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data

represent data on two quantitative variables on a scatter plot and describe how the variables are related

fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models

fit a linear function for a scatter plot that suggests a linear association using given or collected bivariate data

determine and 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. For instance, by looking at a scatterplot, students should be able to tell if the correlation coefficient is positive or negative and give a reasonable estimate of the "r" value. After calculating the line of best fit using technology, students should be able to describe how strong the goodness of fit of the regression is, using "r".

distinguish between correlation and causation

understand that a function from one set (the input, called the domain) to another set (the output, called the range) assigns to each element of the domain exactly one element of the range; each input value maps to exactly one output value. [(e.g., 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).)]

evaluate functions for inputs in their domains using function notation 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 key features of graphs and tables for a function that models a relationship between two quantities in terms of the quantities for a function that models a relationship between two quantities, and sketch graphs showing key features given a verbal description of the relationship (Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior)

relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. e.g., if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function

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

graph functions expressed algebraically and show key features of the graph both by hand and by using technology

graph linear and quadratic functions and show intercepts, maxima, and minima (as determined by the function or by context)

graph exponential functions showing intercepts and end behavior

compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions) (e.g., given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum)

write a function that describes a relationship between two quantities

determine an explicit expression, a recursive process, or steps for calculation from a context

write arithmetic and geometric sequences both recursively and with an explicit formula; use them to model situations, and translate between the two forms. Connect arithmetic sequences to linear functions and geometric sequences to exponential functions

identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them

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. This can be shown by algebraic proof, with a table showing differences, or by calculating average rates of change 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)

show using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or more generally as a polynomial function

interpret the parameters in a linear or exponential function in terms of a context. In context, students should describe what these parameters mean in terms of change and starting value.

write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function

use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of context. For example, compare and contrast quadratic functions in standard, vertex, and intercept forms.

use units of measure (linear, area, capacity, rates, and time) as a way to understand problems. identify, use, and record appropriate units of emasure within context, within data displays, and on graphs. Convert units and rates using dimensional analysis (English to English and Metric to Metric without conversion factor provided and between English and Metric with conversion factor). use units within multistep problems and formulas; interpret units of input and resulting units of output

determine appropriate quantities for the purpose of descriptive modeling. Given a situation, context, or problem, students will determine, identify, and use appropriate quantities for representing the situation.

choose a level of accuracy appropriate to limitations on measurement when reporting quantities. For example, money situations are generally reported to the nearest cent (hundredth). Also, an answers' precision is limited to the precision of the data given.

Rewrite expressions involving radicals and rational exponents using the properties of exponents

Explain why the sum or the product of rational numbers is rational and an irrational number is irrational; and why the product of a nonzero rational number and an irrational number is irrational