Algebra I (Massachusetts Curriculum Frameworks)

The Real Number System (AI.N-RN)

• AI.N-RN.A.1 Explain how rational exponents get their meaning by extending the rules for integer exponents.
• AI.N-RN.A.2 Use exponent rules to rewrite expressions with radicals and rational exponents.
• AI.N-RN.B.3 Explain why sums and products of rationals are rational, and why rational + irrational and non-zero rational × irrational are irrational.

Quantities (AI.N-Q)

• AI.N-Q.A.1 Use units to understand problems and guide multi-step solutions; choose and interpret units and scales in formulas and graphs.
• AI.N-Q.A.2 Define suitable quantities for descriptive modeling.
• AI.N-Q.A.3 Choose a level of accuracy suited to the limits of measurement when reporting quantities.

Seeing Structure in Expressions (AI.A-SSE)

• AI.A-SSE.A.1 Interpret expressions that stand for a quantity in context.
  • a. Explain what terms, factors and coefficients in an expression mean.
  • b. Read a complicated expression by seeing parts of it as single quantities, e.g. P(1 + r)ⁿ as P times a factor.
• AI.A-SSE.A.2 Use the structure of an expression to find ways to rewrite it, e.g. spotting a difference of squares.
• AI.A-SSE.B.3 Rewrite an expression in a form that reveals properties of the quantity it represents.
  • a. Factor a quadratic expression to show the zeros of its function.
  • b. Complete the square to show the maximum or minimum of a quadratic function.
  • c. Use properties of exponents to rewrite exponential expressions, e.g. an annual rate as a monthly rate.

Arithmetic with Polynomials and Rational Expressions (AI.A-APR)

• AI.A-APR.A.1 Understand that polynomials form a system like the integers, closed under certain operations.
  • a. Add, subtract and multiply polynomials and compare the polynomial system with the integers.
  • b. Factor or expand polynomials, collect like terms and use the distributive property.

Creating Equations (AI.A-CED)

• AI.A-CED.A.1 Write and solve one-variable equations and inequalities from linear, quadratic and exponential (integer exponent) situations.
• AI.A-CED.A.2 Write equations in two or more variables for relationships and graph them on labeled, scaled axes.
• AI.A-CED.A.3 Show constraints with equations, inequalities or systems and judge whether solutions make sense in context.
• AI.A-CED.A.4 Rearrange formulas to make a chosen quantity the subject, e.g. Ohm's law V = IR for R.

Reasoning with Equations and Inequalities (AI.A-REI)

• AI.A-REI.A.1 Explain each step in solving an equation as following from equality in the step before, and justify the method.
• AI.A-REI.B.3 Solve one-variable linear equations and inequalities, including ones with letter coefficients.
  • a. Solve one-variable linear equations and inequalities that involve absolute value.
• AI.A-REI.B.4 Solve quadratic equations in one variable.
  • a. Complete the square to turn any quadratic into (x − p)² = q and derive the quadratic formula from it.
  • b. Solve quadratics by inspection, square roots, completing the square, the formula or factoring; write non-real solutions as a ± bi.
• AI.A-REI.C.5 Prove that replacing one equation in a system by itself plus a multiple of the other gives a system with the same solutions.
• AI.A-REI.C.6 Solve pairs of linear equations in two variables exactly and approximately (e.g. by graphing).
• AI.A-REI.C.7 Solve a two-variable system made of a linear and a quadratic equation, by algebra and by graphing.
• AI.A-REI.D.10 Know a two-variable equation's graph shows all of its solutions plotted in the coordinate plane.
• AI.A-REI.D.11 Explain why x-values where y = f(x) and y = g(x) cross solve f(x) = g(x); find solutions approximately.
• AI.A-REI.D.12 Graph linear inequalities in two variables as half-planes and systems of them as overlapping half-planes.

Interpreting Functions (AI.F-IF)

• AI.F-IF.A.1 Understand a function as assigning each input in the domain exactly one output; the graph of f is y = f(x).
• AI.F-IF.A.2 Use function notation, evaluate functions for given inputs and interpret statements in context.
• AI.F-IF.A.3 Know sequences are functions on the integers, sometimes defined recursively, e.g. the Fibonacci sequence.
• AI.F-IF.B.4 Interpret key features of graphs and tables of a relationship in context and sketch graphs from verbal descriptions.
• AI.F-IF.B.5 Relate a function's domain to its graph and to the situation it describes.
• AI.F-IF.B.6 Find, interpret and estimate a function's average rate of change over an interval.
• AI.F-IF.C.7 Graph functions from their equations and show key features, by hand in simple cases and with technology.
  • a. Graph linear and quadratic functions showing intercepts, maxima and minima.
  • b. Graph piecewise functions, including step and absolute value functions.
  • e. Graph exponential functions, marking intercepts and long-run behavior.
• AI.F-IF.C.8 Rewrite a function in equivalent forms to show different properties.
  • a. Factor and complete the square in quadratics to show zeros, extreme values and symmetry.
  • b. Use properties of exponents to read growth or decay rates in exponential functions.
• AI.F-IF.C.9 Move between representations of functions and compare two functions shown in different ways.

Building Functions (AI.F-BF)

• AI.F-BF.A.1 Write linear, quadratic or exponential functions to describe how two quantities are related.
  • a. Find an explicit expression, recursive process or calculation steps from a context.
  • b. Combine standard function types with arithmetic operations.
• AI.F-BF.A.2 Write arithmetic and geometric sequences recursively and with an explicit formula and convert between forms.
• AI.F-BF.B.3 Find the effect of f(x) + k, k·f(x), f(kx) and f(x + k) on graphs of linear, quadratic, exponential and absolute value functions, using technology.
• AI.F-BF.B.4 Find inverse functions using algebra and graphs.
  • a. Solve f(x) = c for an invertible linear function and write the inverse.

Linear, Quadratic, and Exponential Models (AI.F-LE)

• AI.F-LE.A.1 Tell apart situations modeled by linear functions and by exponential functions.
  • a. Prove linear functions grow by equal differences and exponential functions by equal factors over equal intervals.
  • b. Spot situations where one quantity changes by the same amount for each unit of another.
  • c. Spot situations where a quantity grows or shrinks by the same percent over each equal interval.
• AI.F-LE.A.2 Build linear and exponential functions, including sequences, from a graph, a description or input-output pairs.
• AI.F-LE.A.3 Use graphs and tables to show exponential growth eventually exceeds linear, quadratic or polynomial growth.
• AI.F-LE.B.5 Interpret the parameters of a linear or exponential function in context.

Interpreting Categorical and Quantitative Data (AI.S-ID)

• AI.S-ID.A.1 Show data for one variable on the real number line with dot plots, histograms and box plots.
• AI.S-ID.A.2 Compare center and spread of two or more data sets using statistics that suit the shape of the data.
• AI.S-ID.A.3 Explain differences in shape, center and spread of data sets in context, considering outliers.
• AI.S-ID.B.5 Summarize two categorical variables in two-way tables, interpret relative frequencies and spot possible associations.
• AI.S-ID.B.6 Plot pairs of quantitative data on a scatter plot and describe the relationship.
  • a. Fit a function (often linear or exponential) to data and use it to solve problems in context.
  • b. Judge the fit of a function informally by plotting and analyzing residuals.
  • c. Fit a line to a scatter plot that shows a linear pattern.
• AI.S-ID.C.7 Explain what a linear model's slope and intercept mean for the data.
• AI.S-ID.C.8 Use technology to find the correlation coefficient of a linear fit and interpret it.
• AI.S-ID.C.9 Tell correlation apart from causation.