California Common Core Content Standards: Mathematics I
Skills available for California Mathematics I math standards
Domain
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N
Number and Quantity
N-Q Quantities
Reason quantitatively and use units to solve problems.
Reason quantitatively and use units to solve problems.
N-Q.1: 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.
Customary and metric units
Word problems
N-Q.1: 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.
Customary and metric units
Word problems
N-Q.2: Define appropriate quantities for the purpose of descriptive modeling.
Linear models
Exponential models
N-Q.2: Define appropriate quantities for the purpose of descriptive modeling.
Linear models
Exponential models
N-Q.3: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Precision
Greatest possible error
Percent error
N-Q.3: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Precision
Greatest possible error
Percent error
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A
Algebra
A-SSE Seeing Structure in Expressions
Interpret the structure of expressions
Interpret the structure of expressions
A-SSE.1: Interpret expressions that represent a quantity in terms of its context.
A-SSE.1: Interpret expressions that represent a quantity in terms of its context.
A-SSE.1.a: Interpret parts of an expression, such as terms, factors, and coefficients.
A-SSE.1.a: Interpret parts of an expression, such as terms, factors, and coefficients.
A-SSE.1.b: Interpret complicated expressions by viewing one or more of their parts as a single entity.
A-SSE.1.b: Interpret complicated expressions by viewing one or more of their parts as a single entity.
A-CED Creating Equations
Create equations that describe numbers or relationships
Create equations that describe numbers or relationships
A-CED.1: Create equations and inequalities in one variable including ones with absolute value and use them to solve problems.
Linear equations
Absolute value equations
Inequalities
Word problems
A-CED.1: Create equations and inequalities in one variable including ones with absolute value and use them to solve problems.
Linear equations
Absolute value equations
Inequalities
Word problems
A-CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Direct variation
Slope-intercept form
Standard form
Point-slope form
Exponential equations
Linear and exponential equations
A-CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Direct variation
Slope-intercept form
Standard form
Point-slope form
Exponential equations
Linear and exponential equations
A-CED.3: 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.
Linear equations
Linear inequalities
Systems of equations
A-CED.3: 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.
Linear equations
Linear inequalities
Systems of equations
A-CED.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
A-CED.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
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A-REI Reasoning with Equations and Inequalities
Understand solving equations as a process of reasoning and explain the reasoning
Understand solving equations as a process of reasoning and explain the reasoning
A-REI.1: 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.
A-REI.1: 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 equations and inequalities in one variable
A-REI.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Multi-variable equations
Equations with models
One- and two-step linear equations
Multi-step linear equations
Solutions to linear equations
Linear inequalities
Compound inequalities
A-REI.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Multi-variable equations
Equations with models
One- and two-step linear equations
Multi-step linear equations
Solutions to linear equations
Linear inequalities
Compound inequalities
A-REI.3.1: Solve one-variable equations and inequalities involving absolute value, graphing the solutions and interpreting them in context.
A-REI.3.1: Solve one-variable equations and inequalities involving absolute value, graphing the solutions and interpreting them in context.
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Solve systems of equations
Solve systems of equations
A-REI.5: 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.
A-REI.5: 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.
A-REI.6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
Understand solutions
Graphing
Substitution
Elimination
Any method
A-REI.6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
Understand solutions
Graphing
Substitution
Elimination
Any method
Represent and solve equations and inequalities graphically
Represent and solve equations and inequalities graphically
A-REI.10: 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).
A-REI.10: 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).
A-REI.11: 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, polynomial, rational, absolute value, exponential, and logarithmic functions.
Tables
Graphs
A-REI.11: 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, polynomial, rational, absolute value, exponential, and logarithmic functions.
Tables
Graphs
A-REI.12: 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.
A-REI.12: 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.
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F
Functions
F-IF Interpreting Functions
Understand the concept of a function and use function notation
Understand the concept of a function and use function notation
F-IF.1: 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).
Independent and dependent variables
Identify functions
Function tables and graphs
Domain and range
F-IF.1: 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).
Independent and dependent variables
Identify functions
Function tables and graphs
Domain and range
F-IF.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
Evaluate a function
Interpret function notation
Tables and graphs
F-IF.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
Evaluate a function
Interpret function notation
Tables and graphs
F-IF.3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
Arithmetic sequences
Geometric sequences
Recursive formulas
Explicit and recursive sequences
Mixed sequences
F-IF.3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
Arithmetic sequences
Geometric sequences
Recursive formulas
Explicit and recursive sequences
Mixed sequences
Interpret functions that arise in applications in terms of the context
Interpret functions that arise in applications in terms of the context
F-IF.4: 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.
Linear functions
Exponential functions
Linear and exponential functions
Word problems
F-IF.4: 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.
Linear functions
Exponential functions
Linear and exponential functions
Word problems
F-IF.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
F-IF.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
F-IF.6: 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.
F-IF.6: 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.
Analyze functions using different representations
Analyze functions using different representations
F-IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
F-IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
F-IF.7.a: Graph linear and quadratic functions and show intercepts, maxima, and minima.
F-IF.7.a: Graph linear and quadratic functions and show intercepts, maxima, and minima.
F-IF.7.e: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
F-IF.7.e: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
F-IF.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
F-IF.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
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F-BF Building Functions
Build a function that models a relationship between two quantities
Build a function that models a relationship between two quantities
F-BF.1: Write a function that describes a relationship between two quantities.
F-BF.1: Write a function that describes a relationship between two quantities.
F-BF.1.a: Determine an explicit expression, a recursive process, or steps for calculation from a context.
Sequences
Linear and exponential functions
F-BF.1.a: Determine an explicit expression, a recursive process, or steps for calculation from a context.
Sequences
Linear and exponential functions
F-BF.1.b: Combine standard function types using arithmetic operations.
F-BF.1.b: Combine standard function types using arithmetic operations.
F-BF.2: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
F-BF.2: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
Build new functions from existing functions
Build new functions from existing functions
F-BF.3: 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.
F-BF.3: 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.
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F-LE Linear, Quadratic, and Exponential Models
Construct and compare linear, quadratic, and exponential models and solve problems
Construct and compare linear, quadratic, and exponential models and solve problems
F-LE.1: Distinguish between situations that can be modeled with linear functions and with exponential functions.
F-LE.1: Distinguish between situations that can be modeled with linear functions and with exponential functions.
F-LE.1.a: Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
F-LE.1.a: Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
F-LE.1.b: Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
F-LE.1.b: Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
F-LE.1.c: Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
F-LE.1.c: Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
F-LE.2: 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).
Arithmetic and geometric sequences
Slope-intercept form
Point-slope form
Exponential functions
Linear and exponential functions
F-LE.2: 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).
Arithmetic and geometric sequences
Slope-intercept form
Point-slope form
Exponential functions
Linear and exponential functions
F-LE.3: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
F-LE.3: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
Interpret expressions for functions in terms of the situation they model
Interpret expressions for functions in terms of the situation they model
F-LE.5: Interpret the parameters in a linear or exponential function in terms of a context.
F-LE.5: Interpret the parameters in a linear or exponential function in terms of a context.
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G
Geometry
G-CO Congruence
Experiment with transformations in the plane
Experiment with transformations in the plane
G-CO.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
G-CO.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
G-CO.2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
Translations
Reflections
Rotations
Dilations
Classify and compare transformations
G-CO.2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
Translations
Reflections
Rotations
Dilations
Classify and compare transformations
G-CO.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
G-CO.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
G-CO.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
G-CO.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
G-CO.5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Graph an image
Sequences of transformations
G-CO.5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Graph an image
Sequences of transformations
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Understand congruence in terms of rigid motions
Understand congruence in terms of rigid motions
G-CO.6: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
G-CO.6: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
G-CO.7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
G-CO.7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
G-CO.8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
G-CO.8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
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Make geometric constructions
Make geometric constructions
G-CO.12: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).
Angles
Points, lines, and segments
Regular polygons
G-CO.12: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).
Angles
Points, lines, and segments
Regular polygons
G-CO.13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
G-CO.13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
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G-GPE Expressing Geometric Properties with Equations
Use coordinates to prove simple geometric theorems algebraically
Use coordinates to prove simple geometric theorems algebraically
G-GPE.4: Use coordinates to prove simple geometric theorems algebraically.
G-GPE.4: Use coordinates to prove simple geometric theorems algebraically.
G-GPE.5: 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).
G-GPE.5: 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).
G-GPE.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
G-GPE.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
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S
Statistics and Probability
S-ID Interpreting Categorical and Quantitative Data
Summarize, represent, and interpret data on a single count or measurement variable
Summarize, represent, and interpret data on a single count or measurement variable
S-ID.1: Represent data with plots on the real number line (dot plots, histograms, and box plots).
Line plots
Histograms
Bar graphs, line graphs, and histograms
Box plots
S-ID.1: Represent data with plots on the real number line (dot plots, histograms, and box plots).
Line plots
Histograms
Bar graphs, line graphs, and histograms
Box plots
S-ID.2: 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.
S-ID.2: 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.
S-ID.3: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
S-ID.3: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
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Summarize, represent, and interpret data on two categorical and quantitative variables
Summarize, represent, and interpret data on two categorical and quantitative variables
S-ID.5: 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.
S-ID.5: 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.
S-ID.6: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
S-ID.6: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
S-ID.6.a: Fit a function to the data; use functions fitted to data to solve problems in the context of the data.
S-ID.6.a: Fit a function to the data; use functions fitted to data to solve problems in the context of the data.
S-ID.6.b: Informally assess the fit of a function by plotting and analyzing residuals.
S-ID.6.b: Informally assess the fit of a function by plotting and analyzing residuals.
S-ID.6.c: Fit a linear function for a scatter plot that suggests a linear association.
S-ID.6.c: Fit a linear function for a scatter plot that suggests a linear association.
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Interpret linear models
Interpret linear models
S-ID.7: Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
S-ID.7: Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
S-ID.8: Compute (using technology) and interpret the correlation coefficient of a linear fit.
S-ID.8: Compute (using technology) and interpret the correlation coefficient of a linear fit.
S-ID.9: Distinguish between correlation and causation.
S-ID.9: Distinguish between correlation and causation.
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