North Dakota Algebra 2 Math Standards

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N Number and Quantity

• N.RN The Real Number System
  • Extend the properties of exponents to rational numbers
    • N.RN.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.
    • N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
  • Use properties of rational and irrational numbers
    • N.RN.3 Demonstrate that 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.
    • N.RN.4 Perform basic operations on radicals and simplify radicals to write equivalent expressions.
• N.Q Quantities
  • Reason quantitatively and use units to solve problems
    • N.Q.1.i Use units as a way to understand problems and to guide the solution of multi-step problems (e.g., unit analysis).
    • N.Q.1.ii Choose and interpret units consistently in formulas.
    • N.Q.1.iii Choose and interpret the scale and the origin in graphs and data displays.
    • N.Q.2 Define appropriate quantities for the purpose of descriptive modeling.
    • N.Q.3 Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.
• N.CN The Complex Number System
  • Perform arithmetic operations with complex numbers
    • N.CN.1.i Know there is an imaginary number i, such that i² = –1, and every complex number has the form a + bi where a and b are real.
    • N.CN.1.ii Understand the hierarchical relationships among subsets of the complex number system.
    • N.CN.2 Use the definition i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
    • N.CN.3 Use conjugates to find quotients of complex numbers.
  • Use complex numbers in polynomial identities and equations
    • N.CN.7 Solve quadratic equations with real coefficients that have complex solutions.
    • N.CN.8 Extend polynomial identities to the complex numbers.
    • N.CN.9.i Apply the Fundamental Theorem of Algebra to determine the number of zeros for polynomial functions.
    • N.CN.9.ii Find all solutions to a polynomial equation.

A Algebra

• A.SSE Seeing Structure in Expressions
  • Interpret the structure of expressions
    • 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.b Interpret complicated expressions by examining one or more of their parts as a single entity.
    • A.SSE.2 Use the structure of an expression to identify ways to rewrite it.
  • Write expressions in equivalent forms to solve problems
    • A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
      • A.SSE.3.a Factor a quadratic expression to reveal the zeros of the function it defines.
      • A.SSE.3.b Complete the square in a quadratic expression to produce an equivalent expression.
      • A.SSE.3.c Use the properties of exponents to transform exponential expressions.
• A.APR Arithmetic with Polynomials and Rational Expressions
  • Perform arithmetic operations on polynomials
    • A.APR.1.i Add, subtract, and multiply polynomials.
    • A.APR.1.ii Understand that polynomials form a system comparable to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication.
  • Understand the relationship between zeros and factors of polynomials
    • A.APR.2 Apply the Remainder Theorem.
    • A.APR.3.i Identify zeros of polynomials when suitable factorizations are available.
    • A.APR.3.ii Use the zeros to construct a rough graph of the function defined by the polynomial.
  • Use polynomial identities to solve problems
    • A.APR.5 Apply the Binomial Theorem for the expansion of (x + y)ⁿ in powers of x and y for a positive integer n.
  • Rewrite rational expressions
    • A.APR.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
    • A.APR.7.i Add, subtract, multiply, and divide rational expressions.
    • A.APR.7.ii Understand that rational expressions form a system comparable to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression.
• A.CED Creating Equations and Inequalities
  • Create equations that describe numbers or relationships
    • A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
    • A.CED.2.i Create equations in two or more variables to represent relationships between quantities.
    • A.CED.2.ii Graph equations on coordinate axes with appropriate labels and scales.
    • 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.
    • A.CED.4 Rearrange formulas to isolate a quantity of interest, using the same reasoning as in solving equations.
• A.REI Reasoning with Equations and Inequalities
  • Understand solving equations as a process of reasoning and explain the reasoning
    • A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
  • Solve equations and inequalities in one variable
    • A.REI.4 Solve quadratic equations in one variable.
      • A.REI.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.
      • A.REI.4.b Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a + bi for real numbers a and b.
  • Solve systems of equations
    • A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
  • Represent and solve equations and inequalities graphically
    • A.REI.11 Using graphs, technology, tables, or successive approximations, show that the solution(s) to the equation f(x) = g(x) are the x-value(s) that result in the y-values of f(x) and g(x) being the same.

F Functions

• F.IF Interpreting Functions
  • Interpret functions that arise in applications in terms of the context
    • F.IF.4 Use tables, graphs, verbal descriptions, and equations to interpret and sketch the key features of a function modeling the relationship between two quantities.
    • 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.
  • 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.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
      • F.IF.7.c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
      • F.IF.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior.
      • F.IF.7.f Graph f(x) = sin x and f(x)= cos x as representations of periodic phenomena.
    • F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
      • F.IF.8.a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
      • F.IF.8.b Use the properties of exponents to interpret expressions for exponential functions.
    • F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
• F.BF Building Functions
  • 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.b Combine standard function types using arithmetic operations.
      • F.BF.1.c Compose functions.
    • F.BF.2.i Write arithmetic and geometric sequences both recursively and with an explicit formula and convert between the two forms.
    • F.BF.2.ii Use sequences to model situations.
  • Build new functions from existing functions
    • F.BF.3.i Identify the effect on the graph of replacing f(x) by f(x) + k, f(x + k), kf(x), and f(kx), for specific values of k (both positive and negative); find the value of k given the graphs.
    • F.BF.3.ii Recognize even and odd functions from their graphs.
    • F.BF.4 Find inverse functions.
      • F.BF.4.a Write an equation for the inverse given a function has an inverse.
      • F.BF.4.b Verify by composition that one function is the inverse of another.
      • F.BF.4.c Read values of an inverse function from a graph or a table, given that the function has an inverse.
      • F.BF.4.d Produce an invertible function from a non-invertible function by restricting the domain.
    • F.BF.5 Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
• F.LE Linear, Quadratic, and Exponential Models
  • Construct and compare linear, quadratic, and exponential models and solve problems
    • F.LE.4 Use logarithms to express the solution to ab to the ct power = d where a, c, and d are real numbers and b is a positive real number. Evaluate the logarithm using technology when appropriate.
• F.TF Trigonometric Functions
  • Extend the domain of trigonometric functions using the unit circle
    • F.TF.2.i Extend right triangle trigonometry to the four quadrants.
    • F.TF.2.ii Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
    • F.TF.3.i Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6.
    • F.TF.3.ii Use the unit circle to express the values of sine, cosine, and tangent for π – x, π + x, and 2π – x, in terms of their values for x, where x is any real number.
  • Prove and apply trigonometric identities
    • F.TF.8 Prove the Pythagorean identity sin²(Θ) + cos²(Θ)=1 and use it to find n(Θ), cos(Θ), or tan(Θ) given sin(Θ), cos(Θ), or tan(Θ) and the quadrant of the angle.

G Geometry

• G.GPE Expressing Geometric Properties with Equations
  • Understand and use conic sections
    • G.GPE.1.i Derive the equation of a circle of given center and radius.
    • G.GPE.1.ii Derive the equation of a parabola given a focus and directrix.
    • G.GPE.1.iii Derive the equations of ellipses and hyperbolas given foci, using the fact that the sum or difference of distances from the foci is constant.
    • G.GPE.2 Convert between the standard and general form equations of conic sections.
    • G.GPE.3.i Identify key features of conic sections given their equations.
    • G.GPE.3.ii Apply properties of conic sections in real world situations.

S Statistics and Probability

• S.ID Interpreting Categorical and Quantitative Data
  • Summarize, represent, and interpret data on a single count or measurement variable
    • S.ID.4.i Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate.
    • S.ID.4.ii Use calculators, spreadsheets, or tables to estimate areas under the normal curve.
  • Summarize, represent, and interpret data on two categorical and quantitative variables
    • 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 (with or without technology). 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.IC Making Inferences and Justifying Conclusions
  • Understand and evaluate random processes underlying statistical experiments
    • S.IC.1 Understand the process of making inferences about population parameters based on a random sample from that population.
    • S.IC.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.
  • Make inferences and justify conclusions from sample surveys, experiments, and observational studies
    • S.IC.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
    • S.IC.6 Evaluate reports based on data.
      • S.IC.6.a Evaluate articles, reports or websites based on data published in the media by identifying the source of the data, the design of the study, and the way the data are analyzed and displayed.
      • S.IC.6.b Identify and explain misleading use of data; recognize when claims based on data confuse correlation and causation.
      • S.IC.6.c Recognize and describe how graphs and data can be distorted to support different points of view.