The Common Core in Maryland

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Skills available for Maryland high school math standards

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Maryland College and Career-Ready Frameworks: Algebra I Maryland College and Career-Ready Frameworks: Algebra I
Maryland College and Career-Ready Frameworks: Algebra II Maryland College and Career-Ready Frameworks: Algebra II
Maryland College and Career-Ready Frameworks: Geometry Maryland College and Career-Ready Frameworks: Geometry
Maryland College and Career-Ready Standards: Algebra 1 Maryland College and Career-Ready Standards: Algebra 1
Maryland College and Career-Ready Standards: Algebra 2 Maryland College and Career-Ready Standards: Algebra 2
Maryland College and Career-Ready Standards: All High School Pathways Maryland College and Career-Ready Standards: All High School Pathways
Maryland College and Career-Ready Standards: Geometry Maryland College and Career-Ready Standards: Geometry
Maryland College and Career-Ready Standards: Mathematical Practices Maryland College and Career-Ready Standards: Mathematical Practices
Maryland College and Career-Ready Standards: Mathematics I Maryland College and Career-Ready Standards: Mathematics I
Maryland College and Career-Ready Standards: Mathematics II Maryland College and Career-Ready Standards: Mathematics II
Maryland College and Career-Ready Standards: Mathematics III Maryland College and Career-Ready Standards: Mathematics III
Maryland College and Career-Ready Standards: Precalculus A Maryland College and Career-Ready Standards: Precalculus A
Maryland College and Career-Ready Standards: Precalculus B Maryland College and Career-Ready Standards: Precalculus B
Maryland College and Career-Ready Standards: Statistics Maryland College and Career-Ready Standards: Statistics

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1 Polynomial, Rational, and Radical Relationships

Extend the properties of exponents to rational exponents.
- 1.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.
  - 1.N.RN.1.a Ability to use prior knowledge of properties of integer exponents to build understanding of rational exponents and radicals
- 1.N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
  - 1.N.RN.2.a Knowledge of the connection between radical and exponential notation
  - 1.N.RN.2.b Ability to translate between radical and exponential notation
Reason quantitatively and use units to solve problems.
- 1.N.Q.2 Define appropriate quantities for the purpose of descriptive modeling.
  - 1.N.Q.2.a Ability to select and use units of measure to accurately model a given real world scenario
Perform arithmetic operations with complex numbers.
- 1.N.CN.1 Know there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b real.
  - 1.N.CN.1.a Ability to extend experience with solving quadratic equations with no real solution from Algebra I to the existence of complex numbers (e.g. use solving x² + 1 = 0 as a way to introduce complex numbers)
    - Introduction to complex numbers (A2-K.1)
- 1.N.CN.2 Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
  - 1.N.CN.2.a Knowledge of conjugate pairs and the nature of their products
    - Multiply complex numbers (A2-K.4)
    - Multiply and divide complex numbers (PC-V.5)
Use complex numbers in polynomial identities and equations.
- 1.N.CN.7 Solve quadratic equations with real coefficients that have complex solutions.
  - 1.N.CN.7.a Ability to use the quadratic formula and/or completing the square as a means of solving a quadratic equation
  - 1.N.CN.7.b Knowledge that complex solutions occur in conjugate pairs
  - 1.N.CN.7.c Ability to connect experience with solving quadratic equations from Algebra I to situations where analyzing the discriminant will reveal the nature of the solutions which would include complex solutions
- 1.N.CN.8 Extend polynomial identities to the complex numbers.
  - 1.N.CN.8.a Knowledge that a negative number can be thought of as the square of an imaginary number (e.g. -4 = (-2i)²)
- 1.N.CN.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
  - 1.N.CN.9.a Knowledge of the connection between the number of roots and the degree of the polynomial; considering multiple roots, complex roots and distinct real roots
    - Fundamental Theorem of Algebra (A2-Q.8)
    - Fundamental Theorem of Algebra (PC-F.7)
Interpret the structure of expressions.
- 1.A.SSE.2 Use the structure of an expression to identify ways to rewrite it.
  - 1.A.SSE.2.a Ability to use properties of mathematics to alter the structure of an expression
  - 1.A.SSE.2.b Ability to select and then use an appropriate factoring technique
  - 1.A.SSE.2.c Ability to factor expressions completely over complex numbers
Write expressions in equivalent forms to solve problems.
- 1.A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
  - 1.A.SSE.3.c Use the properties of exponents to transform expressions for exponential functions.
    - 1.A.SSE.3.c.1 Ability to connect experience with properties of exponents from Unit 4 of Algebra I to more complex expressions
- 1.A.SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.
  - 1.A.SSE.4.a Knowledge of the difference between an infinite and a finite series
  - 1.A.SSE.4.b Ability to apply the formula for the sum of a finite geometric series: Sn = a(1 - rⁿ)/(1 - r)
    - Find the sum of an arithmetic series (A2-NN.3)
    - Find the sum of a finite geometric series (PC-BB.4)
Understand the relationship between zeros and factors of polynomials.
- 1.A.APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
  - 1.A.APR.2.a Ability to make connections between factors, roots and evaluating functions
  - 1.A.APR.2.b Ability to use both long division and synthetic division
  - 1.A.APR.2.c Ability to use the graph of a polynomial to assist in the efficiency of the process for complicated cases
- 1.A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
  - 1.A.APR.3.a Knowledge of the differences in the end behavior of the graphs as dictated by the leading coefficient and whether the function is even or odd
  - 1.A.APR.3.b Ability to capture the graphical behavior of polynomial functions which have roots with multiplicity greater than one
Use polynomial identities to solve problems.
- 1.A.APR.4 Prove polynomial identities and use them to describe numerical relationships.
  - 1.A.APR.4.a Knowledge of the process for proving identities
  - 1.A.APR.4.b Ability to see, use and manipulate the structure in an expression as needed to prove an identity
- 1.A.APR.5 Know and apply the Binomial Theorem for the expansion of (x + y)ⁿ in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.
  - 1.A.APR.5.a Ability to replicate Pascal's triangle
    - Pascal's triangle (A2-L.8)
    - Pascal's triangle (PC-E.9)
Rewrite rational expressions.
- 1.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.
  - 1.A.APR.6.a Ability to make connections to the Remainder Theorem
- 1.A.APR.7 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
  - 1.A.APR.7.a Ability to make connections between the algorithms for operations on rational numbers and operations on rational expressions
Understand solving equations as a process of reasoning and explain the reasoning.
- 1.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.
  - 1.A.REI.1.a Ability to identify the mathematic al property (addition property of equality, distributive property, etc.) used at each step in the solution process as a means of justifying a step
    - Properties of equality (A1-C.4)
    - Solve linear equations: complete the solution (A1-C.14)
- 1.A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
  - 1.A.REI.2.a Ability to connect prior experience with solving simple equations in one variable to solving equations which require new strategies and additional steps
  - 1.A.REI.2.b Ability to make connections between the domain of a function and extraneous solutions
  - 1.A.REI.2.c Ability to identify extraneous solutions
Solve equations and inequalities in one variable.
- 1.A.REI.4 Solve quadratic equations in one variable.
  - 1.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 reveals that the quadratic equation has "no real solutions".
    - 1.A.REI.4.b.1 Ability to solve quadratic equations using various methods and recognize the most efficient method
    - 1.A.REI.4.b.2 Ability to use the value of the discriminant to determine if a quadratic equation has one double solution, two unique solutions or no real solutions
Solve systems of equations.
- 1.A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
  - 1.A.REI.6.a Ability to extend experiences with solving simultaneous linear equations from 8EE.8 b&c to include solving systems of three equations three unknowns
  - 1.A.REI.6.b Ability to solve systems using the most efficient method
- 1.A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
  - 1.A.REI.7.a Knowledge of the algebraic and graphic representations of quadratic relations as well as quadratic functions
    - Solve a system of linear and quadratic equations (A1-Z.10)
    - Solve a nonlinear system of equations (A2-P.4)
Represent and solve equations and inequalities graphically.
- 1.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.
  - 1.A.REI.11.a Ability to connect experience with solving systems of equations graphically from Algebra I to solving systems that include polynomial, exponential, rational, root, absolute value and logarithmic functions
    - Solve a system of linear and quadratic equations (A1-Z.10)
    - Solve a nonlinear system of equations (A2-P.4)
  - 1.A.REI.11.b Ability to show the equality of two functions using multiple representations
Analyze functions using different representations.
- 1.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.
  - 1.F.IF.7.c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
    - 1.F.IF.7.c.1 Ability to connect experience with graphing linear, exponential and quadratic functions from Algebra I to graphing polynomial functions
      - Graph a quadratic function (A2-N.5)
      - Graph a quadratic function (PC-D.3)
    - 1.F.IF.7.c.2 Ability to identify key features of a function: max, min, intercepts, zeros, and end behaviors.
      - Match polynomials and graphs (A2-Q.11)
      - Match polynomials and graphs (PC-F.10)
Translate between the geometric description and the equation for a conic section.
- 1.G.GPE.2 Derive the equation of a parabola given a focus and directrix.
  - 1.G.GPE.2.a Ability to connect the distance formula and the definition of a parabola
    - Write equations of parabolas in vertex form using properties (A2-DD.6)
  - 1.G.GPE.2.b Ability to connect the algebraic and geometric definitions of a parabola

2 Trigonometric Functions

Extend the domain of trigonometric functions using the unit circle.
- 2.F.TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
  - 2.F.TF.1.a Knowledge that angle measures in radians may be determined by a ratio of intercepted arc to radius
    - Radians and arc length (A2-HH.2)
    - Radians and arc length (PC-R.2)
  - 2.F.TF.1.b Ability to convert between degree and radian measure
    - Convert between radians and degrees (A2-HH.1)
    - Convert between radians and degrees (PC-R.1)
- 2.F.TF.2 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.
  - 2.F.TF.2.a Ability to connect knowledge of special right triangles gained in Geometry to evaluating trigonometric functions at any domain value
  - 2.F.TF.2.b Ability to extend to angles beyond [-2π, 2π], using counterclockwise as the positive direction of rotation
Model periodic phenomena with trigonometric functions.
- 2.F.TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
  - 2.F.TF.5.a Ability to connect contextual situations to appropriate trigonometric function: e.g. using sine or cosine to model cyclical behavior
Prove and apply trigonometric identities.
- 2.F.TF.8 Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
  - 2.F.TF.8.a Ability to make connections to angles in standard position
Summarize, represent, and interpret data on two categorical and quantitative variables.
- 2.S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
  - 2.S.ID.6.a Fit a function to the data; use functions fitted to data to solve problems in the context of the data.
    - 2.S.ID.6.a.1 Ability to recognize types of relationships that lend themselves to linear and exponential models
    - 2.S.ID.6.a.2 Ability to create and use regression models to represent a contextual situation

3 Modeling with Functions

Write expressions in equivalent forms to solve problems.
- 3.A.SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.
  - Find the sum of a finite geometric series (A2-NN.4)
  - Find the sum of a finite geometric series (PC-BB.4)
  - 3.A.SSE.4.a Knowledge of the difference between an infinite and a finite series
  - 3.A.SSE.4.b Ability to apply the formula for the sum of a finite geometric series: S? = a(1–rⁿ)/(1–r)
    - Find the sum of an arithmetic series (A2-NN.3)
    - Find the sum of a finite geometric series (PC-BB.4)
Create equations that describe numbers or relationships.
- 3.A.CED.1 Create equations and inequalities in one variable and use them to solve problems.
  - 3.A.CED.1.a Ability to connect experience from Algebra I with creating linear, exponential and quadratic equations in one variable to include creating simple rational functions
  - 3.A.CED.1.b Ability to distinguish between linear, quadratic, exponential, root and simple rational relationships given the verbal, numeric and/or graphic representations
  - Limitations: In this course rational functions are limited to those whose numerators are of degree at most one and denominators of degree at most 2.
Understand the concept of a function and use function notation.
- 3.F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
Interpret functions that arise in applications in terms of the context.
- 3.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.
  - 3.F.IF.4.a Ability to connect appropriate function to context
  - 3.F.IF.4.b Knowledge of the key features of linear, exponential, polynomial, root, absolute value, piece?wise, simple rational, logarithmic and trigonometric functions
- 3.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.
  - 3.F.IF.6.a Ability to apply this skill to linear, quadratic, polynomial, root and simple rational functions
Analyze functions using different representations.
- 3.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.
  - 3.F.IF.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
    - 3.F.IF.7.e.1 Ability to connect experience with graphing linear and quadratic functions from Algebra I to graphing exponential and logarithmic functions
      - Match exponential functions and graphs II (A1-V.4)
      - Match exponential functions and graphs (A2-Z.5)
    - 3.F.IF.7.e.2 Ability to produce a rough graph of the parent function for each type of function
    - 3.F.IF.7.e.3 Knowledge of how parameters introduced into a function alter the shape of the graph of the parent function
- 3.F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
  - 3.F.IF.8.a Ability to connect experience with writing linear, quadratic and exponential functions in various forms from Algebra I to writing all functions in various forms
  - 3.F.IF.8.b Ability to recognize functions in various forms
  - 3.F.IF.8.b Use the properties of exponents to interpret expressions for exponential functions.
    - 3.F.IF.8.b.1 Ability to connect experience with properties of exponents from Algebra I Unit 2 Linear and Exponential Relationships to more complex expressions.
- 3.F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
  - 3.F.IF.9.a Ability to recognize common attributes of functions from various representations
Build a function that models a relationship between two quantities.
- 3.F.BF.1 Write a function that describes a relationship between two quantities.
  - 3.F.BF.1.a Determine an explicit expression, a recursive process, or steps for calculation from a context.
    - 3.F.BF.1.a.1 Ability to connect experience with linear and exponential functions from Algebra I Unit 2 to quadratic functions
    - 3.F.BF.1.a.2 Ability to write the algebraic representation of a quadratic function from a contextual situation
      - Write a quadratic function from its zeros (A2-N.10)
  - 3.F.BF.1.b Combine standard function types using arithmetic operations.
    - 3.F.BF.1.b.1 Ability to connect experience with adding, subtracting, multiplying and dividing linear, quadratic and exponential functions from Algebra I to adding, subtracting, multiplying and dividing any functions
    - 3.F.BF.1.b.2 Ability to create a new function that is the composition of two or more functions
- 3.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.
- 3.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. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
  - 3.F.BF.3.a Ability to connect experience with this standard as it relates to linear, quadratic and exponential functions from Algebra I to all functions studied
  - 3.F.BF.3.b Ability to make generalizations about the changes that will take place in the graph of any function as a result of making a particular change to the algebraic representation of the function
    - Describe function transformations (A2-W.6)
    - Describe function transformations (PC-C.6)
- 3.F.BF.4 Find inverse functions.
  - 3.F.BF.4.a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.
    - 3.F.BF.4.a.1 Ability to connect experience with finding the inverse of a linear function from Algebra I to finding the inverse of simple exponential, root and rational functions
    - 3.F.BF.4.a.2 Knowledge of the connection of the domain and range of a function to its inverse
    - 3.F.BF.4.a.3 Ability to determine if a function has an inverse
Construct and compare linear, quadratic, and exponential models and solve problems.
- 3.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).
  - 3.F.LE.2.a Ability to produce an algebraic model
- 3.F.LE.4 For exponential models, express as a logarithm the solution to ab to the ct power = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
  - 3.F.LE.4.a Knowledge that logarithmic functions are inverses of exponential functions
  - 3.F.LE.4.b Knowledge of the properties of logarithms and exponents and their connection to one another
Interpret expressions for functions in terms of the situation they model.
- 3.F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context.
  - 3.F.LE.5.a Ability to interpret the slope and y-intercept of a linear model in terms of context
    - Solve one-step and two-step linear equations: word problems (A1-C.9)
    - Solve linear equations: word problems (A2-B.3)
  - 3.F.LE.5.b Ability to identify the initial amount present in an exponential model (f(0)=b⁰ + k = 1 + k)
  - 3.F.LE.5.c Ability to interpret the rate of increase/decrease in an exponential model

4 Inferences and Conclusions from Data

Summarize, represent, and interpret data on a single count or measurement variable.
- 4.S.ID.4 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. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
  - 4.S.ID.4.a Ability to construct, interpret and use normal curves, based on standard deviation
  - 4.S.ID.4.b Ability to identify data sets as approximately normal or not
  - 4.S.ID.4.c Ability to estimate and interpret area under curves using the Empirical Rule (68?95?98.7%)
Understand and evaluate random processes underlying statistical experiments.
- 4.S.IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
  - 4.S.IC.1.a Knowledge of various sampling methods (e.g., simple random, convenience, stratified…)
  - 4.S.IC.1.b Ability to select an appropriate sampling technique for a given situation
  - 4.S.IC.1.c Ability to explain in context the difference between values describing a population and a sample
- 4.S.IC.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.
  - 4.S.IC.2.a Ability to calculate and analyze theoretical and experimental probabilities accurately
    - Theoretical probability (A1-KK.1)
    - Experimental probability (A1-KK.2)
  - 4.S.IC.2.b Knowledge of various types of sampling procedures and ability to select and carry out the appropriate process for a given situation
  - 4.S.IC.2.c Ability to design, conduct and interpret the results of simulations
  - 4.S.IC.2.d Ability to explain and use the Law of Large Numbers
Make inferences and justify conclusions from sample surveys, experiments, and observational studies.
- 4.S.IC.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
  - 4.S.IC.3.a Ability to conduct sample surveys, experiments and observational studies
  - 4.S.IC.3.b Understanding of the limitations of observational studies that do not allow major conclusions on treatments
    - Experiment design (A2-RR.8)
    - Experiment design (PC-FF.8)
  - 4.S.IC.3.c Ability to recognize and avoid bias
- 4.S.IC.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
  - 4.S.IC.4.a Ability to informally establish bounds as to when something is statistically significant
  - 4.S.IC.4.b Ability to conduct simulations and accurately interpret and use the results
    - Experimental probability (A1-KK.2)
  - 4.S.IC.4.c Ability to use sample means and sample proportions to estimate population values
    - Find confidence intervals for population means (PC-FF.5)
    - Find confidence intervals for population proportions (PC-FF.6)
- 4.S.IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
  - 4.S.IC.5.a Ability to set up and conduct a randomized experiment or investigation, collect data and interpret the results
  - 4.S.IC.5.b Ability to draw conclusions based on comparisons of simulation versus experimental results
    - Analyze the results of an experiment using simulations (PC-FF.9)
  - 4.S.IC.5.c Ability to determine the statistical significance of data
- 4.S.IC.6 Evaluate reports based on data.
Use probability to evaluate outcomes of decisions.
- 4.S.MD.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
- 4.S.MD.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
  - 4.S.MD.7.a Ability to synthesize and apply various probability concepts to evaluate decisions
    - Choose the better bet (A2-PP.10)
    - Choose the better bet (PC-DD.10)

5 Applications of Probability

Understand independence and conditional probability and use them to interpret data.
- 5.S.CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").
  - 5.S.CP.1.a Ability to describe a sample space
  - 5.S.CP.1.b Understanding of and ability to use set notation, key vocabulary and graphic organizers linked to this standard
- 5.S.CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
  - 5.S.CP.2.a Ability to determine the conditional probability of an event given that another event occurs
  - 5.S.CP.2.b Ability to determine the probability of an event given the probability of a complementary event
    - Calculate probabilities of events (A2-OO.1)
    - Calculate probabilities of events (PC-CC.1)
  - 5.S.CP.2.c Ability to determine if two events are dependent or independent
- 5.S.CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
  - 5.S.CP.3.a Understanding of and ability to use set notation, key vocabulary and graphic organizers linked to this standard
- 5.S.CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.
  - 5.S.CP.4.a Ability to connect experience with two-way frequency tables from Algebra I to sample spaces
    - Find probabilities using two-way frequency tables (A2-OO.5)
    - Find probabilities using two-way frequency tables (PC-CC.4)
  - 5.S.CP.4.b Knowledge of the characteristics of conditional probability
    - Find conditional probabilities using two-way frequency tables (A2-OO.10)
    - Find conditional probabilities using two-way frequency tables (PC-CC.8)
- 5.S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
  - 5.S.CP.5.a Ability to make connections between statistical concepts and real world situations
Use the rules of probability to compute probabilities of compound events in a uniform probability model.
- 5.S.CP.6 Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model.
  - 5.S.CP.6.a Ability to analyze a situation to determine the conditional probability of a described event given that another event occurs
    - Geometric probability (G-AA.14)
- 5.S.CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
  - 5.S.CP.7.a Ability to analyze a situation to determine the conditional probability of a described event given that another event occurs
    - Find probabilities using the addition rule (A2-OO.11)
    - Find probabilities using the addition rule (PC-CC.9)
  - 5.S.CP.7.b Ability to make connections between numeric results and context
    - Find probabilities using the addition rule (A2-OO.11)
    - Find probabilities using the addition rule (PC-CC.9)
- 5.S.CP.8 Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
  - 5.S.CP.8.a Ability to analyze a situation to determine the probability of a described event
    - Find conditional probabilities (A2-OO.8)
    - Find conditional probabilities (PC-CC.6)
  - 5.S.CP.8.b Ability to make connections between numeric results and context
- 5.S.CP.9 Use permutations and combinations to compute probabilities of compound events and solve problems.
  - 5.S.CP.9.a Ability to use formulas containing factorial notation
  - 5.S.CP.9.b Ability to analyze a situation to determine the probability of a described event
    - Find probabilities using combinations and permutations (A2-OO.4)
    - Find probabilities using combinations and permutations (PC-CC.3)
Use probability to evaluate outcomes of decisions.
- 5.S.MD.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
  - 5.S.MD.6.a Ability make connections between the numeric probabilities and context
  - 5.S.MD.6.b Knowledge of the Law of Large Numbers
- 5.S.MD.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
  - 5.S.MD.7.a Knowledge of and ability to use a variety of data collection techniques
  - 5.S.MD.7.b Ability make connections between the numeric probabilities and context
    - Choose the better bet (A2-PP.10)
    - Choose the better bet (PC-DD.10)
  - 5.S.MD.7.c Knowledge of the Law of Large Numbers

The Common Core in Maryland

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1 Polynomial, Rational, and Radical Relationships

Extend the properties of exponents to rational exponents.

1.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.

1.N.RN.1.a Ability to use prior knowledge of properties of integer exponents to build understanding of rational exponents and radicals

1.N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

1.N.RN.2.a Knowledge of the connection between radical and exponential notation

1.N.RN.2.b Ability to translate between radical and exponential notation

Reason quantitatively and use units to solve problems.

1.N.Q.2 Define appropriate quantities for the purpose of descriptive modeling.

1.N.Q.2.a Ability to select and use units of measure to accurately model a given real world scenario

Perform arithmetic operations with complex numbers.

1.N.CN.1 Know there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b real.

1.N.CN.1.a Ability to extend experience with solving quadratic equations with no real solution from Algebra I to the existence of complex numbers (e.g. use solving x² + 1 = 0 as a way to introduce complex numbers)

1.N.CN.2 Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

1.N.CN.2.a Knowledge of conjugate pairs and the nature of their products

Use complex numbers in polynomial identities and equations.

1.N.CN.7 Solve quadratic equations with real coefficients that have complex solutions.

1.N.CN.7.a Ability to use the quadratic formula and/or completing the square as a means of solving a quadratic equation

1.N.CN.7.b Knowledge that complex solutions occur in conjugate pairs

1.N.CN.7.c Ability to connect experience with solving quadratic equations from Algebra I to situations where analyzing the discriminant will reveal the nature of the solutions which would include complex solutions

1.N.CN.8 Extend polynomial identities to the complex numbers.

1.N.CN.8.a Knowledge that a negative number can be thought of as the square of an imaginary number (e.g. -4 = (-2i)²)

1.N.CN.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

1.N.CN.9.a Knowledge of the connection between the number of roots and the degree of the polynomial; considering multiple roots, complex roots and distinct real roots

Interpret the structure of expressions.

1.A.SSE.2 Use the structure of an expression to identify ways to rewrite it.

1.A.SSE.2.a Ability to use properties of mathematics to alter the structure of an expression

1.A.SSE.2.b Ability to select and then use an appropriate factoring technique

1.A.SSE.2.c Ability to factor expressions completely over complex numbers

Write expressions in equivalent forms to solve problems.

1.A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

1.A.SSE.3.c Use the properties of exponents to transform expressions for exponential functions.

1.A.SSE.3.c.1 Ability to connect experience with properties of exponents from Unit 4 of Algebra I to more complex expressions

1.A.SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.

1.A.SSE.4.a Knowledge of the difference between an infinite and a finite series

1.A.SSE.4.b Ability to apply the formula for the sum of a finite geometric series: Sn = a(1 - rn)/(1 - r)

Understand the relationship between zeros and factors of polynomials.

1.A.APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

1.A.APR.2.a Ability to make connections between factors, roots and evaluating functions

1.A.APR.2.b Ability to use both long division and synthetic division

1.A.APR.2.c Ability to use the graph of a polynomial to assist in the efficiency of the process for complicated cases

1.A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

1.A.APR.3.a Knowledge of the differences in the end behavior of the graphs as dictated by the leading coefficient and whether the function is even or odd

1.A.APR.3.b Ability to capture the graphical behavior of polynomial functions which have roots with multiplicity greater than one

Use polynomial identities to solve problems.

1.A.APR.4 Prove polynomial identities and use them to describe numerical relationships.

1.A.APR.4.a Knowledge of the process for proving identities

1.A.APR.4.b Ability to see, use and manipulate the structure in an expression as needed to prove an identity

1.A.APR.5 Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.

1.A.APR.5.a Ability to replicate Pascal's triangle

Rewrite rational expressions.

1.A.APR.6.a Ability to make connections to the Remainder Theorem

1.A.APR.7 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

1.A.APR.7.a Ability to make connections between the algorithms for operations on rational numbers and operations on rational expressions

Understand solving equations as a process of reasoning and explain the reasoning.

1.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.

1.A.REI.1.a Ability to identify the mathematic al property (addition property of equality, distributive property, etc.) used at each step in the solution process as a means of justifying a step

1.A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

1.A.REI.2.a Ability to connect prior experience with solving simple equations in one variable to solving equations which require new strategies and additional steps

1.A.REI.2.b Ability to make connections between the domain of a function and extraneous solutions

1.A.REI.2.c Ability to identify extraneous solutions

Solve equations and inequalities in one variable.

1.A.REI.4 Solve quadratic equations in one variable.

1.A.REI.4.b.1 Ability to solve quadratic equations using various methods and recognize the most efficient method

1.A.REI.4.b.2 Ability to use the value of the discriminant to determine if a quadratic equation has one double solution, two unique solutions or no real solutions

Solve systems of equations.

1.A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

1.A.REI.6.a Ability to extend experiences with solving simultaneous linear equations from 8EE.8 b&c to include solving systems of three equations three unknowns

1.A.REI.6.b Ability to solve systems using the most efficient method

1.A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

1.A.REI.7.a Knowledge of the algebraic and graphic representations of quadratic relations as well as quadratic functions

Represent and solve equations and inequalities graphically.

1.A.REI.11.a Ability to connect experience with solving systems of equations graphically from Algebra I to solving systems that include polynomial, exponential, rational, root, absolute value and logarithmic functions

1.A.REI.11.b Ability to show the equality of two functions using multiple representations

Analyze functions using different representations.

1.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.

1.F.IF.7.c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

1.A.SSE.4.b Ability to apply the formula for the sum of a finite geometric series: Sn = a(1 - rⁿ)/(1 - r)

1.A.APR.5 Know and apply the Binomial Theorem for the expansion of (x + y)ⁿ in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.

3.A.SSE.4.b Ability to apply the formula for the sum of a finite geometric series: S? = a(1–rⁿ)/(1–r)