Algebra 1

Number and Quantity

Algebra

Functions

Statistics and Probability

The Real Number System

Quantities

Seeing Structure in Expressions

Arithmetic with Polynomials and Rational Expressions

Creating Equations

Reasoning with Equations and Inequalities

Interpreting Functions

Building Functions

Linear, Quadratic, and Exponential Models

Summarize, represent, and interpret data on a single count or measurement variable.

Conditional Probability and the rules of Probability

Use properties of rational and irrational numbers.

Reason quantitatively and use units to solve problems. 

Interpret the structure of expressions.

Write expressions in equivalent forms to solve problems.

Perform arithmetic operations on polynomials. 

Understand the relationship between zeros and factors of polynomials.

Create equations that describe numbers or relationships.

Understand solving equations as a process of reasoning and explain the reasoning.

Solve equations and inequalities in one variable. 

Solve systems of equations. 

Represent and solve equations and inequalities graphically.

Understand the concept of a function and use function notation.

Interpret functions that arise in applications in terms of the context

Analyze functions using different representations.

Build a function that models a relationship between two quantities.

Build new functions from existing functions.

Construct and compare linear, quadratic, and exponential models and solve problems.

Interpret expressions for functions in terms of the situation they model.

Summarize, represent, and interpret data on a single count or measurement variable.

Summarize, represent, and interpret data on two categorical and quantitative variables.

Interpret linear models.

Understand independence and conditional probability and use them to interpret data.

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

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, include utilizing real-world context.

Define appropriate quantities for the purpose of descriptive modeling. Include problem-solving opportunities utilizing real-world context.

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities utilizing realworld context.

Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret expressions by viewing one or more of their parts as a single entity. 

Use structure to identify ways to rewrite numerical and polynomial expressions. Focus on polynomial multiplication and factoring patterns. 

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

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.

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Focus on quadratic and cubic polynomials in which linear and quadratic factors are available. 

Create equations and inequalities in one variable and use them to solve problems. Include problem-solving opportunities utilizing real-world context. Focus on linear, quadratic, exponential and piecewise-defined functions (limited to absolute value and step).

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.

Explain each step in solving linear and quadratic equations 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 linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – k) 2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection (e.g., x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Focus on solutions for quadratic equations that have real roots. Include cases that recognize when a quadratic equation has no real solutions.

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 systems of linear equations exactly and approximately, focusing on pairs of linear equations in two variables. Include problem solving opportunities utilizing real-world context.

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). Focus on cases where f(x) and/or g(x) are linear, quadratic, exponential and piecewise-defined functions (limited to absolute value and step).

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.

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).

Evaluate a function for inputs in the domain, 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.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Include problem-solving opportunities utilizing real-world context. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums. Focus on linear, quadratic, exponential and piecewise-defined functions (limited to absolute value and step).

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 continuous function (presented symbolically or as a table) on a closed interval. Estimate the rate of change from a graph. Include problem-solving opportunities utilizing real-world context. Focus on linear, quadratic, exponential and piecewise-defined functions (limited to absolute value and step).

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Focus on linear, quadratic, exponential and piecewise-defined functions (limited to absolute value and step).

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square of a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Focus on linear, quadratic, exponential and piecewise-defined functions (limited to absolute value and step).

Write a function that describes a relationship between two quantities. Determine an explicit expression, a recursive process, or steps for calculation from real-world context. Focus on linear, quadratic, exponential and piecewise-defined functions (limited to absolute value and step). 

Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), 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. Focus on linear, quadratic, exponential and piecewise-defined functions (limited to absolute value and step).

Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. 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 input/output pairs.

Observe, using graphs and tables, that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically.

Interpret the parameters in a linear or exponential function with integer exponents utilizing real world context.

Represent real-value data with plots for the purpose of comparing two or more data sets. 

Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) 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 outliers if present.

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 quantities are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Focus on linear models. b. Informally assess the fit of a function by plotting and analyzing residuals.

Interpret the slope as a rate of change and the constant term of a linear model in the context of the data.

Compute and interpret the correlation coefficient of a linear relationship.

Distinguish between correlation and causation. 

Describe events as subsets of a sample space using characteristics of the outcomes, or as unions, intersections, or complements of other events.

Use the Multiplication Rule for independent events to understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

Explain the pattern for the sum or product for combinations of rational and irrational numbers.

Determine the necessary unit(s) to use to solve real world problems.

Solve real world problems involving units of measurement.

No CCC developed for this standard.

Describe the accuracy of measurement when reporting quantity (you can lessen your limitations by measuring precisely) 

Translate an algebraic expression into a word problem.

Simplify expressions that include exponents.

Rewrite expressions that include rational exponents.

Rewrite mathematical statements (e.g., an expression) in multiple forms.

No CCC developed for this standard.

No CCC developed for this standard. 

Translate a real-world problem into a one variable linear equation.

Solve equations with one or two variables using equations or graphs

No CCC developed for this standard.

No CCC developed for this standard. 

Solve equations with one or two variables using equations or graphs

Solve equations with one or two variables using equations or graphs

Solve a linear equation to find a missing attribute given the area, surface area, or volume and the other attribute.

No CCC developed for this standard.

No CCC developed for this standard.

No CCC developed for this standard.

No CCC developed for this standard.

No CCC developed for this standard.

No CCC developed for this standard.

No CCC developed for this standard.

No CCC developed for this standard.

No CCC developed for this standard.

No CCC developed for this standard.

No CCC developed for this standard.

No CCC developed for this standard.

No CCC developed for this standard.

No CCC developed for this standard.

No CCC developed for this standard.

No CCC developed for this standard.

No CCC developed for this standard.

Select the appropriate graphical representation of a linear model based on real world events.

In a linear situation using graphs or numbers, predicts the change in rate based on a given change in one variable (e.g., If I have been adding sugar at a rate of 1T per cup of water, what happens to my rate if I switch to 2T of sugar for every cup of water?).

No CCC developed for this standard.

No CCC developed for this standard.

No CCC developed for this standard.

Complete a graph given the data, using dot plots, histograms, or box plots.

Compare means, median, and range of 2 sets of data.

No CCC developed for this standard.

Design study using categorical and continuous data, including creating a question, identifying a sample, and making a plan for data collection.

Use descriptive statistics: range, median, mode, mean, outliers/gaps to describe the data set.

Design study using categorical and continuous data, including creating a question, identifying a sample, and making a plan for data collection.

Use descriptive statistics: range, median, mode, mean, outliers/gaps to describe the data set.

Interpret the rate of change using graphical representations.

No CCC developed for this standard.

No CCC developed for this standard.

No CCC developed for this standard.

No CCC developed for this standard.