# Secondary Mathematics Essential Standards

## School Year 2020-21

For the FCPS Return to School in 2020-21, the standards listed below emphasize the central priorities for learning in each course. These standards are intended to guide instruction and assessment during this extraordinary school year.

## Honors

Unit guides provided to teachers contain recommended extensions for students enrolled in the honors course option.

## Year Long Standards

These standards are the central priorities for learning in the course. They are prioritized in planning, teaching, assessing, and intervening to ensure student success over the course of the year.

### Math 7

- Build on the concept of ratios and rational numbers to solve problems including problems that require proportional reasoning.
- Develop an understanding of and fluency with solving linear equations and inequalities in one variable by applying the properties of real numbers
- Develop an understanding of and fluency with multiple representations of functions that model a multiplicative or additive relationship.

### Math 7 Honors

- Apply understandings of operations with rational numbers to solve practical problems, including consumer problems and percent of change.
- Apply understandings of multiple representations of functions to model a linear relationship, including practical problems; emphasis is placed on rate of change and y-intercept.
- Extend understanding of and fluency in simplifying expressions and solving equations and inequalities with rational numbers, including practical problems.

### Prealgebra

- Apply understandings of operations with rational numbers to solve practical problems, including consumer problems and percent of change.
- Apply understandings of multiple representations of functions to model a linear relationship, including practical problems; emphasis is placed on rate of change and y-intercept.
- Extend understanding of and fluency in simplifying expressions and solving equations and inequalities with rational numbers, including practical problems.

### Algebra 1

- Understand that practical problems can be interpreted, represented, and solved using expressions, equations and inequalities.
- Apply understandings of multiple representations of functions to model linear and quadratic relationships, including practical problems; and to use graphing tools to visualize, analyze and communicate understanding of function behaviors and in justifying and verifying solutions.
- Generalize patterns in manipulating and simplifying algebraic expressions.

### Geometry

- Create geometric models and understand definitions, postulates, and theorems. Attend to and learn how to work with imagery, visualizing, diagramming, and constructing.
- Develop reasoning skills through the exploration of geometric relationships and communicate through proof.
- Represent and solve a variety of practical problems, applying their understanding of geometric attributes while connecting to algebraic reasoning.

### Algebra, Functions, and Data Analysis (AFDA)

- Represent and solve a variety of practical problems that require the formulation of linear, quadratic, exponential, or logarithmic models or a system of equations, using real-world data.
- Use the language and symbols of mathematics in representations and communication, both orally and in writing.
- Use graphing tools to visualize, analyze and communicate understanding of algebraic and statistical behaviors and in justifying and verifying solutions.

### Algebra 2

- Understand and compare the properties of classes of functions, including polynomial, square and cube root, rational, exponential, and logarithmic functions.
- Understand that practical problems can be interpreted, represented, and solved using equations and inequalities.
- Use graphing tools to visualize, analyze and communicate understanding of function behaviors and in justifying and verifying solutions.

### Precalculus

- Make connections amongst the multiple representations of functions (i.e. polynomial, rational, logarithmic, exponential, trigonometric)
- Understand and connect the unit circle to trigonometric function values, trigonometric graphs, and polar functions*
- *Further develop an understanding of equivalence by:
- representing functions in different forms and coordinate systems (conics written as polar and parametric functions)
- juxtaposing trigonometric proofs and trigonometric equations

- Solve and model practical real-world problems involving parametric, polynomial, rational, logarithmic, and exponential functions, trigonometric functions and laws, and vectors
- Develop an understanding of the notion of infinity (continuous, smooth domains) to prepare for calculus by studying limits and continuity, and convergent/divergent series

*Honors

## Quarter 1 (September 8 – October 30)

### Math 7

- Compare and order rational numbers (non-calculator skill).
- Solve practical problems involving operations with rational numbers.
- Solve single-step and multistep practical problems, using proportional reasoning.

### Math 7 Honors

- Determine both the positive and negative square roots of a given perfect square.
- Solve practical problems involving consumer applications
- Determine whether a given relation is a function
- Determine the domain and range of a function.
- Recognize and describe the graph of a linear function with a slope that is positive, negative, or zero.
- Identify the slope and y-intercept of a linear function given a table of values, a graph, or an equation in y = mx + b form.
- Determine the independent and dependent variable, given a practical situation modeled by a linear function;
- Graph a linear function given the equation in y = mx + b form.
- Make connections between and among representations of a linear function using verbal descriptions, tables, equations, and graphs.

### Prealgebra

- Determine both the positive and negative square roots of a given perfect square.
- Solve practical problems involving consumer applications
- Determine whether a given relation is a function
- Determine the domain and range of a function.
- Recognize and describe the graph of a linear function with a slope that is positive, negative, or zero.
- Identify the slope and y-intercept of a linear function given a table of values, a graph, or an equation in y = mx + b form.
- Determine the independent and dependent variable, given a practical situation modeled by a linear function;
- Graph a linear function given the equation in y = mx + b form.
- Make connections between and among representations of a linear function using verbal descriptions, tables, equations, and graphs.

### Algebra 1

- Evaluate algebraic expressions for given replacement values of the variables.
- Solve multistep linear equations in one variable algebraically.
- Solve literal equations for a specified variable.
- Solve multistep linear inequalities in one variable algebraically and represent the solution graphically.
- Investigate and analyze linear and quadratic function families and their characteristics both algebraically and graphically, including:
- Determining whether a relation is a function.
- Domain and range.
- Connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs.

### Geometry

- Solve problems involving symmetry and transformation. This will include:
- Investigating and using formulas for determining distance, midpoint, and slope.
- Applying slope to verify and determine whether lines are parallel or perpendicular.
- Investigating symmetry and determining whether a figure is symmetric with respect to a line or a point.
- Determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods.

- Use the relationships between angles formed by two lines intersected by a transversal to:
- Prove two or more lines are parallel.
- Solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal.

### Algebra, Functions, and Data Analysis (AFDA)

- Investigate and analyze linear, quadratic, exponential, and logarithmic function families and their characteristics. Key concepts include
- Domain and range.
- Intervals on which a function is increasing or decreasing.
- Absolute maxima and minima.
- Zeros.
- Intercepts.
- Values of a function for elements in its domain.
- Connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs.
- End behavior.

- Design and conduct a survey. Key concepts include:
- Sample size.
- Sampling technique.
- Controlling sources of bias.
- Data collection.
- Data analysis and reporting.

### Algebra 2

- For functions [and relations]:
- Use knowledge of transformations to convert between equations and the corresponding graphs of functions.
- Investigate and analyze functions and relations graphically. Key concepts include:
- Domain and range.
- Intervals in which a function is increasing or decreasing.
- Extrema.
- Zeros.
- Intercepts.
- Values of a function for elements in its domain.
- Inverse of a function.

- Factor polynomials completely in one variable.
- Solve quadratic equations over the set of [real] complex numbers.
- For polynomial functions:
- Recognize the general shape of function families.
- Use knowledge of transformations to convert between equations and the corresponding graphs of functions.

- Investigate and analyze quadratic function families algebraically and graphically. Key concepts include
- Domain and range.
- Intervals in which a function is increasing or decreasing.
- Extrema.
- Zeros.
- Intercepts.
- Values of a function for elements in its domain.
- Inverse of a function.

- Investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression.

### Precalculus

- Investigate and identify the properties of polynomial, rational, piecewise, and step functions and sketch the graphs of the functions.
- Identify a polynomial and rational function, given an equation or graph.
- Given a graph or equation of a polynomial or rational function, identify:
- domain and range;
- zeros (real and complex);
- intercepts;
- symmetry;
- asymptotes (horizontal, vertical, and oblique/slant);
- points of discontinuity;
- intervals for which the function is increasing, decreasing or constant;
- end behavior; and
- relative and/or absolute maximum and minimum points.

- Sketch the graph of a polynomial or rational function.
- Investigate and verify characteristics of polynomial and rational functions, using a graphing utility.
- Rationalize the denominator of a rational function.

- Investigate and identify the characteristics of exponential and logarithmic functions to graph the function, solve equations, and solve practical problems.
- Identify exponential functions from an equation or a graph.
- Identify logarithmic functions from an equation or a graph.
- Define e, and know its approximate value.
- Convert between equations written in logarithmic and exponential form.
- Identify common and natural logarithms, given an equation or practical situation.
- Use laws of exponents and logarithms to solve equations and simplify expressions.
- Model practical problems, using exponential and logarithmic functions.
- Graph exponential and logarithmic functions and identify asymptotes, end behavior, intercepts, domain, and range

- Apply compositions of functions and inverses of functions to practical situations and investigate and verify the domain and range of resulting functions.

## Quarter 2 (November 2 – January 22)

### Math 7

- Determine square roots of perfect squares (non-calculator skill)
- Identify and describe absolute value of rational numbers.
- Evaluate algebraic expressions for given replacement values of the variable
- Solve two-step linear equations in one variable, including practical problems that require the solution of a two-step linear equation in one variable.
- Solve one- and two-step linear inequalities in one variable, including practical problems, involving addition, subtraction, multiplication, and division, and graph the solution on a number line.

### Math 7 Honors

- Recognize and describe the graph of a linear function with a slope that is positive, negative, or zero;
- Identify the slope and y-intercept of a linear function given a table of values, a graph, or an equation in y = mx + b form;
- Determine the independent and dependent variable, given a practical situation modeled by a linear function;
- Graph a linear function given the equation in y = mx + b form; and
- Make connections between and among representations of a linear function using verbal descriptions, tables, equations, and graphs.
- Evaluate an algebraic expression for given replacement values of the variables.
- simplify algebraic expressions in one variable.
- Solve multistep linear equations in one variable with the variable on one or both sides of the equation, including practical problems that require the solution of a multistep linear equation in one variable.

### Prealgebra

- Recognize and describe the graph of a linear function with a slope that is positive, negative, or zero;
- Identify the slope and y-intercept of a linear function given a table of values, a graph, or an equation in y = mx + b form;
- Graph a linear function given the equation in y = mx + b form; and
- Evaluate an algebraic expression for given replacement values of the variables.
- Simplify algebraic expressions in one variable.
- Solve multistep linear equations in one variable with the variable on one or both sides of the equation, including practical problems that require the solution of a multistep linear equation in one variable.

### Algebra 1

- Determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line.
- Write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line.
- Graph linear equations in two variables.
- Solve practical problems involving equations and systems of equations.

### Geometry

- Given information concerning the lengths of sides and/or measures of angles in triangles, will solve problems, including practical problems. This will include
- ordering the sides by length, given angle measures;
- ordering the angles by degree measure, given side lengths;
- determining whether a triangle exists; and
- determining the range in which the length of the third side must lie.

- Given information in the form of a figure or statement, will prove two triangles are congruent.
- Given information in the form of a figure or statement, will prove two triangles are similar.
- Apply the concepts of similarity to two- or three-dimensional geometric figures. This will include
- solving problems, including practical problems, about similar geometric figures.

### Algebra, Functions, and Data Analysis (AFDA)

- Investigate and analyze linear function families and their characteristics. Key concepts include
- domain and range;
- intervals on which a function is increasing or decreasing;
- zeros;
- intercepts;
- values of a function for elements in its domain;
- connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs.

- Collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems using models of linear functions.
- Use multiple representations of functions for analysis, interpretation, and prediction.

### Algebra 2

- Solve quadratic equations over the set of complex numbers
- Solve systems of linear-quadratic and quadratic-quadratic graphically.
- Investigate and analyze quadratic function families graphically. Key concepts include
- extrema;
- zeros;
- intercepts;
- values of a function for elements in its domain;
- connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs.

- Collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic functions.
- For polynomial functions:
- recognize the general shape of function families; and
- use knowledge of transformations to convert between equations and the corresponding graphs of functions.

- Investigate and analyze polynomial function families algebraically and graphically. Key concepts include
- domain, range, and continuity;
- intervals in which a function is increasing or decreasing;
- extrema;
- zeros;
- intercepts;
- values of a function for elements in its domain;
- connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; and
- end behavior.

- Investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression.

### Precalculus

- Given a point on the terminal side of an angle in standard position, or the value of the trigonometric function of the angle, will determine the sine, cosine, tangent, cotangent, secant, and cosecant of the angle.
- Develop and apply the properties of the unit circle in degrees and radians.
- Determine the value of any trigonometric function
- Given one of the six trigonometric functions in standard form, will
- state the domain and the range of the function;
- determine the amplitude, period, phase shift, vertical shift, and asymptotes;
- sketch the graph of the function by using transformations for at least a two-period interval; and
- investigate the effect of changing the parameters in a trigonometric function on the graph of the function.

- Verify basic trigonometric identities and make substitutions, using the basic identities.
- reciprocal identities;
- Pythagorean identities;
- sum and difference identities;
- double-angle identities; and
- half-angle identities.

## Quarter 3 (January 25 – March 26)

### Math 7

- Determine the slope, m, as rate of change in a proportional relationship between two quantities and write an equation in the form y = mx to represent the relationship;
- Graph a line representing a proportional relationship between two quantities given the slope and an ordered pair, or given the equation in y =mx form where m represents the slope as rate of change.
- Determine the y-intercept, b, in an additive relationship between two quantities and write an equation in the form y = x + b to represent the relationship;
- Graph a line representing an additive relationship between two quantities given the y-intercept and an ordered pair, or given the equation in the form y = x + b, where b represents the y-intercept; and
- Make connections between and among representations of a proportional or additive relationship between two quantities using verbal descriptions, tables, equations, and graphs.
- Solve problems, including practical problems, involving the relationship between corresponding sides and corresponding angles of similar quadrilaterals and triangles.
- Determine the theoretical and experimental probabilities of an event; and
- Investigate and describe the difference between the experimental probability and theoretical probability of an event

### Math 7 Honors

- Solve multistep linear inequalities in one variable with the variable on one or both sides of the inequality symbol, including practical problems, and graph the solution on a number line.
- Verify and apply the Pythagorean Theorem; and
- Solve area and perimeter problems, including practical problems, involving composite plane figures.

### Prealgebra

- Solve multistep linear inequalities in one variable with the variable on one or both sides of the inequality symbol, including practical problems, and graph the solution on a number line.
- Verify and apply the Pythagorean Theorem; and
- Solve area and perimeter problems, including practical problems, involving composite plane figures.

### Algebra 1

- Perform operations on polynomials, including applying the laws of exponents to perform operations on expressions;
- Simplify
- square roots of whole numbers;
- numerical expressions containing square roots

- Perform operations on polynomials, including
- adding, subtracting, multiplying, and dividing polynomials; and
- factoring completely first- and second-degree binomials and trinomials in one variable.

### Geometry

- Verify and use properties of quadrilaterals to solve problems, including practical problems.
- Solve problems, including practical problems, by applying properties of circles. This will include determining
- Arc length.
- Area of a sector.

- Use surface area and volume of three-dimensional objects to solve practical problems.
- Apply the concepts of similarity to two- or three-dimensional geometric figures.
- Determining how changes in one or more dimensions of a figure affect area and/or volume of the figure;
- Determining how changes in area and/or volume of a figure affect one or more dimensions of the figure

### Algebra, Functions, and Data Analysis (AFDA)

- Investigate and analyze quadratic function families and their characteristics. Key concepts include
- domain and range;
- intervals on which a function is increasing or decreasing;
- zeros;
- intercepts;
- values of a function for elements in its domain;
- connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs.

- Collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems using models of quadratic functions.
- Use multiple representations of functions for analysis, interpretation, and prediction.

### Algebra 2

- Add, subtract, multiply, divide, and simplify radical expressions containing rational numbers and variables, and expressions containing rational exponents.
- Solve equations containing radical expressions.
- For square root and cube root functions
- recognize the general shape of function families; and
- use knowledge of transformations to convert between equations and the corresponding graphs of functions.

- Investigate and analyze square root and cube root function families algebraically and graphically. Key concepts include
- domain and range;
- extrema;
- zeros;
- intercepts;
- values of a function for elements in its domain;
- connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs;
- end behavior; and
- inverse of a function.

- Investigate and analyze rational function families algebraically and graphically. Key concepts include
- domain, range, and continuity;
- intervals in which a function is increasing or decreasing;
- extrema;
- zeros;
- intercepts;
- values of a function for elements in its domain;
- end behavior.

- Add, subtract, multiply, divide, and simplify rational algebraic expressions.
- Solve equations containing rational algebraic expressions
- For rational functions:
- recognize the general shape of function families; and
- use knowledge of transformations to convert between equations and the corresponding graphs of functions.

### Precalculus

- Graph the six inverse trigonometric functions.
- Solve trigonometric equations and inequalities.
- Determine the value of any trigonometric function and inverse trigonometric function
- Create and solve practical problems involving triangles (i.e. Pythagorean Theorem, Law of Sines, and Law of Cosines to solve practical problems).
- Solve problems, including practical problems.

## Quarter 4 (April 5 – June 11)

### Math 7

- Given data in a practical situation, will
- Represent data in a histogram;
- Make observations and inferences about data represented in a histogram

- Describe and determine the volume and surface area of rectangular prisms and cylinders; and
- Solve problems, including practical problems, involving the volume and surface area of rectangular prisms and cylinders.

### Math 7 Honors

- Solve problems, including practical problems, involving volume and surface area of cones and square-based pyramids;
- Compare and contrast the probability of independent and dependent events; and
- Determine probabilities for independent and dependent events.
- Represent data in scatterplots;
- Make observations about data represented in scatterplots; and
- Use a drawing to estimate the line of best fit for data represented in a scatterplot.

### Prealgebra

- Solve problems, including practical problems, involving volume and surface area of cones and square-based pyramids;
- Compare and contrast the probability of independent and dependent events; and
- Determine probabilities for independent and dependent events.
- Represent data in scatterplots;
- Make observations about data represented in scatterplots; and
- Use a drawing to estimate the line of best fit for data represented in a scatterplot.

### Algebra 1

- Solve
- quadratic equations in one variable algebraically;
- practical problems involving equations and systems of equations.

- Investigate and analyze linear and quadratic function families and their characteristics both algebraically and graphically, including
- Determining whether a relation is a function;
- Domain and range;
- Zeros;
- Intercepts;
- Values of a function for elements in its domain; and
- Connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs.

- Collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of linear and quadratic functions.

### Geometry

- Solve problems, including practical problems, involving right triangles. This will include applying the Pythagorean Theorem and its converse.
- Use deductive reasoning to construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include determineing the validity of a logical argument.
- Solve problems, including practical problems, involving right triangles. This will include applying properties of
- special right triangles.
- trigonometric ratios.

### Algebra, Functions, and Data Analysis (AFDA)

- Use knowledge of transformations to write an equation, given the graph of a function (linear, quadratic).
- Calculate probabilities. Key concepts include
- conditional probability;
- dependent and independent events;
- addition and multiplication rules;
- counting techniques (permutations and combinations); and
- Law of Large Numbers.

- Identify and describe properties of a normal distribution;
- Interpret and compare z-scores for normally distributed data; and
- Apply properties of normal distributions to determine probabilities associated with areas under the standard normal curve.

### Algebra 2

- Investigate and analyze exponential and logarithmic function families algebraically and graphically. Key concepts include
- domain, range, and continuity;
- intervals in which a function is increasing or decreasing;
- extrema;
- zeros;
- intercepts;
- values of a function for elements in its domain;
- end behavior.

- For exponential and logarithmic functions:
- recognize the general shape of function families; and

### Precalculus

- Investigate, graph, and identify the properties of conic sections from equations in vertex and standard form.
- Investigate and verify graphs of transformed conic sections, using a graphing utility.
- Graph conic sections from equations written in vertex/standard form using transformations.
- Identify properties of conic sections.

- Use parametric equations to model and solve practical problems.
- Graph parametric equations.
- Use a graphing utility to graph and analyze parametric equations

- Determine the limit of an algebraic function, if it exists, as the variable approaches either a finite number or infinity.
- Verify estimates about the limit of a function using a graphing utility.
- Determine the limit of a function algebraically and verify with a graphing utility.
- Determine the limit of a function numerically and verify with a graphing utility.
- Use limit notation when describing end behavior of a function

- Investigate and describe the continuity of functions.
- Describe continuity of a function.
- Investigate the continuity of functions including absolute value, step, rational, and piecewise functions, using graphical and algebraic methods.
- Classify types of discontinuity.
- Prove continuity at a point, using the definition of limits.