Massachusetts flagMassachusetts: Model Algebra I [AI] Math Standards

61 standards · 10 domains

ALGEBRA - ARITHMETIC WITH POLYNOMIALS AND RATIONAL EXPRESSIONS

  • AI.A-APR.A.1.a Perform operations on polynomial expressions (addition, subtraction, multiplication), and compare the system of polynomials to the system of integers when performing operations.
  • AI.A-APR.A.1.b Factor and/or expand polynomial expressions, identify and combine like terms, and apply the Distributive property.

ALGEBRA - CREATING EQUATIONS

  • AI.A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. (Include equations arising from linear, quadratic, and exponential functions with integer exponents.)
  • AI.A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
  • AI.A-CED.A.3 Represent constraints by linear equations or inequalities, and by systems of linear equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
  • AI.A-CED.A.4 Rearrange formulas to highlight a quantity of interest using the same reasoning as in solving equations (Properties of equality). For example, rearrange Ohm’s law R = V^2/P to solve for voltage, V. Manipulate variables in formulas used in financial contexts such as for simple interest, I=Prt.

ALGEBRA - REASONING WITH EQUATIONS AND INEQUALITIES

  • AI.A-REI.A.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 or refute a solution method.
  • AI.A-REI.B.3.a Solve linear equations and inequalities in one variable involving absolute value.
  • AI.A-REI.B.4.a 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 this form.
  • AI.A-REI.B.4.b Solve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when the solutions of a quadratic equation results in non-real solutions and write them as a ± bi for real numbers a and b.
  • AI.A-REI.C.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.
  • AI.A-REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
  • AI.A-REI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x^2 + y^2 = 3.
  • AI.A-REI.D.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). Show that any point on the graph of an equation in two variables is a solution to the equation.
  • AI.A-REI.D.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 and make tables of values. Include cases where f(x) and/or g(x) are linear and exponential functions.
  • AI.A-REI.D.12 Graph the solutions of 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 of a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

ALGEBRA - SEEING STRUCTURE IN EXPRESSIONS

  • AI.A-SSE.A.1.a Interpret parts of an expression, such as terms, factors, and coefficients.
  • AI.A-SSE.A.1.b Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)^t as the product of P and a factor not depending on P.
  • AI.A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see (x + 2)^2 – 9 as a difference of squares that can be factored as ((x + 2) + 3)((x + 2 ) – 3).
  • AI.A-SSE.B.3.a Factor a quadratic expression to reveal the zeros of the function it defines.
  • AI.A-SSE.B.3.b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
  • AI.A-SSE.B.3.c Use the properties of exponents to transform expressions for exponential functions. For example, the expression 1.15^t can be rewritten as (1.15^(1/12))^12t ≈ 1.012^12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

FUNCTIONS - BUILDING FUNCTIONS

  • AI.F-BF.A.1.a Determine an explicit expression, a recursive process, or steps for calculation from a context.
  • AI.F-BF.A.1.b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
  • AI.F-BF.A.2 Write arithmetic and geometric sequences both recursively and with an explicit formula them to model situations, and translate between the two forms.
  • AI.F-BF.B.3 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. Include linear, quadratic, exponential, and absolute value functions. Utilize technology to experiment with cases and illustrate an explanation of the effects on the graph.
  • AI.F-BF.B.4.a Solve an equation of the form f(x) = c for a linear function f that has an inverse and write an expression for the inverse.

FUNCTIONS - INTERPRETING FUNCTIONS

  • AI.F-IF.A.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 (range) of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
  • AI.F-IF.A.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. For example, given a function representing a car loan, determine the balance of the loan at different points in time.
  • AI.F-IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n + 1) = f(n) + f(n - 1) for n ≥ 1.
  • AI.F-IF.B.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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; and end behavior.
  • AI.F-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, 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.
  • AI.F-IF.B.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.
  • AI.F-IF.C.7.a Graph linear and quadratic functions and show intercepts, maxima, and minima.
  • AI.F-IF.C.7.b Graph piecewise-defined functions, including step functions and absolute value functions.
  • AI.F-IF.C.7.e Graph exponential functions showing intercepts and end behavior.
  • AI.F-IF.C.8.a Use the process of factoring and completing the square in a quadratic function to show zeros, maximum/minimum values, and symmetry of the graph, and interpret these in terms of a context.
  • AI.F-IF.C.8.b Use the properties of exponents to interpret expressions for exponential functions. Apply to financial situations such as identifying appreciation and depreciation rate for the value of a house or car some time after its initial purchase: V_n = P(1+r)^n. For example, identify percent rate of change in functions such as y = (1.02)^t, y = (0.97)^t, y = (1.01)^12t, and y = (1.2)^(t∕10), and classify them as representing exponential growth or decay.
  • AI.F-IF.C.9 Translate among different representations of functions (algebraically, graphically, numerically in tables, or by verbal descriptions). Compare properties of two functions each represented in a different way. For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

FUNCTIONS - LINEAR, QUADRATIC, AND EXPONENTIAL MODELS

  • AI.F-LE.A.1.a Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
  • AI.F-LE.A.1.b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
  • AI.F-LE.A.1.c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
  • AI.F-LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (including reading these from a table).
  • AI.F-LE.A.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.
  • AI.F-LE.B.5 Interpret the parameters in a linear or exponential function (of the form f(x) = b^x + k) in terms of a context.

NUMBER AND QUANTITY - QUANTITIES

  • AI.N-Q.A.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.
  • AI.N-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.
  • AI.N-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

NUMBER AND QUANTITY - THE REAL NUMBER SYSTEM

  • AI.N-RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^1/3 to be the cube root of 5 because we want (5^1/3)^3 = 5^((1/3)3) to hold, so (5^1/3)^3 must equal 5.
  • AI.N-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
  • AI.N-RN.B.3 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.

STATISTICS AND PROBABILITY - INTERPRETING CATEGORICAL AND QUANTITATIVE DATA

  • AI.S-ID.A.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).
  • AI.S-ID.A.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.
  • AI.S-ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
  • AI.S-ID.B.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.
  • AI.S-ID.B.6.a Fit a linear function to the data and use the fitted function to solve problems in the context of the data. Use functions fitted to data or choose a function suggested by the context (emphasize linear and exponential models).
  • AI.S-ID.B.6.b Informally assess the fit of a function by plotting and analyzing residuals.
  • AI.S-ID.B.6.c Fit a linear function for a scatter plot that suggests a linear association.
  • AI.S-ID.C.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
  • AI.S-ID.C.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.
  • AI.S-ID.C.9 Distinguish between correlation and causation.