Algebra I Standards
Test Specifications | Total Point Range: 54-56 (Proficiency: ~35% correct)
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Verify 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.
Verify 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 algebraically and graphically. a. Limit to pairs of linear equations in two variables.
Solve systems of linear equations algebraically and graphically. a. Limit to pairs of linear equations in two variables.
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).
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 equation 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, making tables of values, or finding successive approximations.
Explain why the x-coordinates of the points where the graphs of the equation 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, making tables of values, or finding successive approximations.
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.
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).
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.
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.
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. Key features include the following: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
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. Key features include the following: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
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Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Use the properties of exponents to interpret expressions for exponential functions. i. Focus on exponential functions evaluated at integer inputs.
Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Use the properties of exponents to interpret expressions for exponential functions. i. Focus on exponential functions evaluated at integer inputs.
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
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Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
Identify the effect on the graph of replacing f(x) by f(x) + k, kf(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.
Identify the effect on the graph of replacing f(x) by f(x) + k, kf(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.
Find inverse functions.
Find inverse functions.
a. Informally determine the input of a function when the output is known. (A1, M1)
a. Informally determine the input of a function when the output is known. (A1, M1)
b. Read values of an inverse function from a graph or a table, given that the function has an inverse. (A2, M3)
b. Read values of an inverse function from a graph or a table, given that the function has an inverse. (A2, M3)
c. Verify by composition that one function is the inverse of another. (A2, M3)
c. Verify by composition that one function is the inverse of another. (A2, M3)
d. Find the inverse of a function algebraically, given that the function has an inverse. (A2, M3)
d. Find the inverse of a function algebraically, given that the function has an inverse. (A2, M3)
e. Produce an invertible function from a non-invertible function by restricting the domain.
e. Produce an invertible function from a non-invertible function by restricting the domain.
Distinguish between situations that can be modeled with linear functions and with exponential functions.
Distinguish between situations that can be modeled with linear functions and with exponential functions.
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).
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).
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically.
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 in terms of a context.
Interpret the parameters in a linear or exponential function in terms of a context.
Number, Quantities, Equations and Expressions
Number, Quantities, Equations and Expressions
(Point Range: 19-22, Weight: 33-41%)
(Point Range: 19-22, Weight: 33-41%)
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. ★
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. ★
Define appropriate quantities for the purpose of descriptive modeling.
Define appropriate quantities for the purpose of descriptive modeling.
Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Interpret expressions that represent a quantity in terms of its context.
Interpret expressions that represent a quantity in terms of its context.
Use the structure of an expression to identify ways to rewrite it. For example, to factor 3x(x – 5) + 2(x – 5), students should recognize that the ""x – 5"" is common to both expressions being added, so it simplifies to (3x+2)(x – 5); or see x4– y4 as (x2)2– (y2)2, thus recognizing it as a difference of squares that can be factored as (x2– y2)(x2+y2).
Use the structure of an expression to identify ways to rewrite it. For example, to factor 3x(x – 5) + 2(x – 5), students should recognize that the ""x – 5"" is common to both expressions being added, so it simplifies to (3x+2)(x – 5); or see x4– y4 as (x2)2– (y2)2, thus recognizing it as a difference of squares that can be factored as (x2– y2)(x2+y2).
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
Understand that polynomials form a system analogous to the integers, namely, that they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Understand that polynomials form a system analogous to the integers, namely, that they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
a. Focus on polynomial expressions that simplify to forms that are linear or quadratic.
a. Focus on polynomial expressions that simplify to forms that are linear or quadratic.
Create equations and inequalities in one variable and use them to solve problems. Include equations and inequalities arising from linear, quadratic, simple rational, and exponential functions.
Create equations and inequalities in one variable and use them to solve problems. Include equations and inequalities arising from linear, quadratic, 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.★
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.★
a. Focus on applying linear and simple exponential expressions. (A1, M1)
a. Focus on applying linear and simple exponential expressions. (A1, M1)
b. Focus on applying simple quadratic expressions. (A1, M2)
b. Focus on applying simple quadratic expressions. (A1, M2)
c. Extend to include more complicated function situations with the option to graph with technology. (A2, M3)
c. Extend to include more complicated function situations with the option to graph with technology. (A2, M3)
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.
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.
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
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.
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 linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Solve quadratic equations in one variable.
Solve quadratic equations in one variable.
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
Statistics (Point Range: 10-12, Weight: 18-22%)
Statistics (Point Range: 10-12, Weight: 18-22%)
Represent data with plots on the real number line (dot plotsG, histograms, and box plots) in the context of real-world applications using the GAISE model.
Represent data with plots on the real number line (dot plotsG, histograms, and box plots) in the context of real-world applications using the GAISE model.
In the context of real-world applications by using the GAISE model, use statistics appropriate to the shape of the data distribution to compare center (median and mean) and spread (mean absolute deviationG, interquartile rangeG, and standard deviation) of two or more different data sets.
In the context of real-world applications by using the GAISE model, use statistics appropriate to the shape of the data distribution to compare center (median and mean) and spread (mean absolute deviationG, interquartile rangeG, and standard deviation) of two or more different data sets.
In the context of real-world applications by using the GAISE model, interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
In the context of real-world applications by using the GAISE model, 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 thedata (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of thedata (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.★
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.★
a. 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. (A2, M3)
a. 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. (A2, M3)
b. Informally assess the fit of a function by discussing residuals. (A2, M3)
b. Informally assess the fit of a function by discussing residuals. (A2, M3)
c. Fit a linear function for a scatterplot that suggests a linear association. (A1, M1)
c. Fit a linear function for a scatterplot that suggests a linear association. (A1, M1)
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
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.
Compute (using technology) and interpret the correlation coefficient of a linear fit.
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