Courses of Study : Mathematics

Number Systems and Operations
Together, irrational numbers and rational numbers complete the real number system, representing all points on the number line, while there exist numbers beyond the real numbers called complex numbers.
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
1. Explain how the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for an additional notation for radicals in terms of rational exponents. [Algebra I with Probability, 1]
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Explain how the meaning of rational exponents follows from extending the properties of integer exponents
  • Use rational exponent notation to represent radicals
Teacher Vocabulary:
  • Rational exponent
  • Radical
  • Root Index
  • Radicand
Knowledge:
Students know:
  • The meaning of rational exponents
  • Where the value of a number to a rational exponent derives from, based on a table and graph
Skills:
Students are able to:
  • Explain how the meaning of rational exponents derives from a table, graph, and properties of integer exponents.
Understanding:
Students understand that:
  • the denominator of the rational exponent is the root index and the numerator is the exponent of the radicand.
Diverse Learning Needs:
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. [Algebra I with Probability, 2]
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Convert from radical representation to using rational exponents and vise versa.
Teacher Vocabulary:
  • Radical
  • Rational exponent
  • Expression
Knowledge:
Students know:
  • the denominator of the rational exponent is the root index and the numerator is the exponent of the radicand. For example, 51/2=5 and 163/2=(161/2)3= (16)3=43=64.
  • The root index of the radical is the denominator of the rational exponent and the exponent of the radicand is the numerator of the rational exponent. For example, 4103= 103/4.
Skills:
Students are able to:
  • Rewrite expressions from radical representations to rational exponents and vice versa.
Understanding:
Students understand:
  • The meaning of rational exponents and how to convert between them and radical representations.
Diverse Learning Needs:
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
3. Define the imaginary number i such that i2 = -1. [Algebra I with Probability, 3]
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Define the imaginary number i as the square root of -1.
  • Define i2As equal to -1.
Teacher Vocabulary:
  • Imaginary numbers
  • Real numbers
  • Irrational numbers
  • Rational numbers
  • Complex numbers
Knowledge:
Students know:
  • when a negative number is square rooted on most calculators, the solution gives an error message.
  • The square root of -1 is i.
  • There is another number system, the complex numbers, that include imaginary numbers. -complex numbers look like 5i+2 or 10i. -i2=ii= -1-1=-1.
Skills:
Students are able to:
  • find a square root of a negative number using i.
    For example, -25=5i.
Understanding:
Students understand:
  • how to find the square root of a negative number using i.
  • The difference in the real numbers and complex numbers.
Diverse Learning Needs:
Algebra and Functions
Expressions can be rewritten in equivalent forms by using algebraic properties, including properties of addition, multiplication, and exponentiation, to make different characteristics or features visible
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
4. Interpret linear, quadratic, and exponential expressions in terms of a context by viewing one or more of their parts as a single entity. [Algebra I with Probability, 4]

Example: Interpret the accrued amount of investment P(1 + r)t , where P is the principal and r is the interest rate, as the product of P and a factor depending on time t.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Connect each part of a linear, quadratic, or exponential expression to its corresponding context in a story situation.
  • Explain the meaning of individual parts of the expression such as terms, factors, and coefficients.
Teacher Vocabulary:
  • Expression
  • Terms
  • Coefficient
  • Factors
  • linear expression
  • quadratic expression
  • Exponential expression
Knowledge:
Students know:
  • Interpretations of parts of algebraic expressions such as terms, factors, and coefficients.
Skills:
Students are able to:
  • Produce mathematical expressions that model given contexts.
  • Provide a context that a given mathematical expression accurately fits.
  • Explain the reasoning for selecting a particular algebraic expression by connecting the quantities in the expression to the physical situation that produced them.
Understanding:
Students understand that:
  • Physical situations can be represented by algebraic expressions which combine numbers from the context, variables representing unknown quantities, and operations indicated by the context.
  • Different, but equivalent, algebraic expressions can be formed by approaching the context from a different perspective.
Diverse Learning Needs:
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
5. Use the structure of an expression to identify ways to rewrite it. [Algebra I with Probability, 5]

Example: See x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2).
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Rewrite algebraic expressions by combining like terms or factoring to reveal equivalent forms of the same expression.
Teacher Vocabulary:
  • like terms
  • Expression
  • Factor
  • properties of operations (Appendix D, Table 1)
  • Difference of squares
Knowledge:
Students know:
  • Properties of operations (including those in Appendix D, Table 1),
  • When one form of an algebraic expression is more useful than an equivalent form of that same expression.
Skills:
Students are able to:
  • -Use algebraic properties to produce equivalent forms of the same expression by recognizing underlying mathematical structures.
    For example, 3(x-5) = 3x-15 and 2a+12 = 2(a+6) or 3a-a+10+2and x2-2x-15 = (x-5) (x+3).
Understanding:
Students understand that:
  • Generating simpler, but equivalent, algebraic expressions facilitates the investigation of more complex algebraic expressions.
Diverse Learning Needs:
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
6. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

a. Factor quadratic expressions with leading coefficients of one, and use the factored form to reveal the zeros of the function it defines.

b. Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines; complete the square to find the vertex form of quadratics with a leading coefficient of one.

c. Use the properties of exponents to transform expressions for exponential functions. [Algebra I with Probability, 6]

Example: Identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, or y = (1.2)t/10, and classify them as representing exponential growth or decay.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Make sense of algebraic expressions by identifying structures within the expression which allow them to rewrite it in useful ways that assist in the solution of given problems
  • Factor a quadratic expression with leading coefficient of one to reveal the zeros of the function it defines
  • Complete the square in a quadratic expression to reveal the maximum or minimum value, the axis of symmetry, and the vertex form of the quadratic expression.
  • Justify their selection of a form for an expression by explaining which features of the expression are revealed by the particular form and how these features aid in resolving a problem situation.
  • Apply exponential properties to expressions and explain and justify the meaning in a contextual situation.
Teacher Vocabulary:
  • Function
  • zero of a function
  • Roots
  • parabola
  • vertex form of a quadratic expression
  • Minimum and maximum value
  • Axis of symmetry
  • Completing the square
  • Exponential growth and decay
Knowledge:
Students know:
  • The vertex form of a quadratic expression as f (x) = a(x
  • h)2 + k, where (h, k) is the vertex of the parabola.
  • Techniques for generating equivalent forms of an algebraic expression including factoring and completing the square for quadratic expressions and using properties of exponents,
  • When one form of an algebraic expression is more useful than an equivalent form of that same expression to solve a given problem.
Skills:
Students are able to:
  • Use algebraic properties including properties of exponents to produce equivalent forms of the same expression by recognizing underlying mathematical structures,
  • Factor quadratic expressions with leading coefficient of one
  • Complete the square in quadratic expressions.
Understanding:
Students understand that:
  • An expression may be written in various equivalent forms.
  • Some forms of the expression are more beneficial for revealing key properties of the function.
Diverse Learning Needs:
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
7. Add, subtract, and multiply polynomials, showing that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication. [Algebra I with Probability, 7]
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Use the repeated reasoning from prior knowledge of properties of arithmetic on integers to progress consistently to rules for arithmetic on polynomials,
  • Accurately perform combinations of operations (addition, subtraction, multiplication) on various polynomials.
Teacher Vocabulary:
  • Polynomials
  • Closure
  • Analogous system
Knowledge:
Students know:
  • Corresponding rules of arithmetic of integers, specifically what it means for the integers to be closed under addition, subtraction, and multiplication, and not under division,
  • Procedures for performing addition, subtraction, and multiplication on polynomials.
Skills:
Students are able to:
  • Communicate the connection between the rules for arithmetic on integers and the corresponding rules for arithmetic on polynomials,
  • Accurately perform combinations of operations on various polynomials.
Understanding:
Students understand that:
  • There is an operational connection between the arithmetic on integers and the arithmetic on polynomials.
Diverse Learning Needs:
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
8. Analyze the relationship (increasing or decreasing, linear or non-linear) between two quantities represented in a graph. [Grade 8, 17]
Unpacked Content
Evidence Of Student Attainment:
Students:
Given graphical representations of functions,
  • use mathematical reasoning to analyze the graphs and describe the functional relationships between two quantities.
Teacher Vocabulary:
  • Increasing
  • Decreasing
  • linear
  • non-linear
Knowledge:
Students know:
  • Characteristics of representations for functions in graphic form.
Skills:
Students are able to:
  • Use mathematical vocabulary and understanding of functions to describe relationships between two quantities.
Understanding:
Students understand that:
  • Functions can be represented in a variety of ways, each of which provides unique perspectives of the relationship between the variables.
  • Graphs of functions are useful to compare characteristics of different relationships.
Diverse Learning Needs:
Analyze and solve linear equations and systems of two linear equations.
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
9. Solve systems of two linear equations in two variables by graphing and substitution.

a. Explain that the solution(s) of systems of two linear equations in two variables corresponds to points of intersection on their graphs because points of intersection satisfy both equations simultaneously.

b. Interpret and justify the results of systems of two linear equations in two variables (one solution, no solution, or infinitely many solutions) when applied to real-world and mathematical problems. [Grade 8, 12]
Unpacked Content
Evidence Of Student Attainment:
Students:
  • graph a system of two linear equations, recognizing that the ordered pair for the point of intersection is the x-value that will generate the given y-value for both equations.
  • Recognize that graphed lines with one point of intersection (different slopes) will have one solution, parallel lines (same slope, different y-intercepts) have no solutions, and lines that are the same (same slope, same y-intercept) will have infinitely many solutions.
  • Use substitution to solve a system, given two linear equations in slope-intercept form or one equation in standard form and one in slope-intercept form.
  • Make sense of their solutions by making connections between algebraic and graphical solutions and the context of the system of linear equations.
Teacher Vocabulary:
  • System of linear equations
  • point of intersection
  • one solution
  • no solution
  • Infinitely many solutions
  • parallel lines
  • Slope-intercept form of a linear equation
  • Standard form of a linear equation
Knowledge:
Students know:
  • The properties of operations and equality and their appropriate application.
  • Graphing techniques for linear equations (using points, using slope-intercept form, using technology).
  • Substitution techniques for algebraically finding the solution to a system of linear equations.
Skills:
Students are able to:
  • generate a table from an equation.
  • Graph linear equations.
  • Identify the ordered pair for the point of intersection.
  • Explain the meaning of the point of intersection ( or lack of intersection point) in context.
  • Solve a system algebraically using substitution when both equations are written in slope-intercept form or one is written in standard form and the other in slope-intercept form.
Understanding:
Students understand that:
  • Any point on a line when substituted into the equation of the line, makes the equation true and therefore, the intersection point of two lines must make both equations true,
  • Graphs and equations of linear relationships are different representations of the same relationships, but reveal different information useful in solving problems, and allow different solution strategies leading to the same solutions.
Diverse Learning Needs:
Finding solutions to an equation, inequality, or system of equations or inequalities requires the checking of candidate solutions, whether generated analytically or graphically, to ensure that solutions are found and that those found are not extraneous.
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
10. Explain why extraneous solutions to an equation involving absolute values may arise and how to check to be sure that a candidate solution satisfies an equation. [Algebra I with Probability, 8]
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Solve equations involving absolute value
  • Decide if a number is a solution to an absolute value equation by substituting the number into the original equation. For
    Example: 2x+12=4x produces solutions of x=6 and x=-2. However -2 is not a solution since when it is substituted back into the original equation it yields an untrue statement of -8=8. x=-2 is an extraneous solution.
Teacher Vocabulary:
  • Extraneous solution
  • Absolute value
Knowledge:
Students know:
  • Absolute value cannot equal a negative number.
  • Substitution techniques to determine true or false statements.
Skills:
Students are able to:
  • solve absolute value equations.
  • Substitute possible solutions into original equations to determine if it is in fact a solution or not.
Understanding:
Students understand that:
  • That not all solutions generated algebraically actually satisfy the original absolute value equation.
Diverse Learning Needs:
The structure of an equation or inequality (including, but not limited to, one-variable linear and quadratic equations, inequalities, and systems of linear equations in two variables) can be purposefully analyzed (with and without technology) to determine an efficient strategy to find a solution, if one exists, and then to justify the solution.
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
11. Select an appropriate method to solve a quadratic equation in one variable.

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. Explain how the quadratic formula is derived from this form.

b. Solve quadratic equations by inspection (such as x2 = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real. [Algebra I with Probability, 9]
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Select an appropriate method (taking square roots, factoring, completing the square, or quadratic formula) for solving a quadratic equation in one variable based on its original form.
  • Use completing the square to transform any quadratic equation into the form (x-p)2=q.
  • Derive the quadratic formula from (x-p)2=q.
  • Recognize that some solutions may not be real or namely that they may be imaginary or complex numbers.
  • Provide reasonable approximations when appropriate in a graph or table.
Teacher Vocabulary:
  • quadratic equation
  • Square root
  • Factoring
  • Completing the square
  • quadratic formula
  • Derive
  • Real numbers
  • Imaginary numbers
  • Complex numbers
Knowledge:
Students know:
  • Any real number has two square roots, that is, if a is the square root of a real number then so is -a.
  • The method for completing the square.
  • A quadratic equation in standard form (ax2+bx+c=0) has real roots when b2-4ac is greater than or equal to zero and complex roots when b2-4ac is less than zero.
Skills:
Students are able to:
  • Take the square root of both sides of an equation.
  • Factor quadratic expressions in the form x2+bx+c where the leading coefficient is one.
  • Use the factored form to find zeros of the function.
  • Complete the square.
  • Use the quadratic formula to find solutions to quadratic equations.
  • Manipulate equations to rewrite them into other forms.
Understanding:
Students understand that:
  • Solutions to a quadratic equation must make the original equation true and this should be verified.
  • When the quadratic equation is derived from a contextual situation, proposed solutions to the quadratic equation should be verified within the context given, as well as mathematically.
  • Different procedures for solving quadratic equations are necessary under different conditions.
  • If ab=0, then at least one of a or b must be zero (a=0 or b=0) and this is then used to produce the two solutions to the quadratic equation.
  • Whether the roots of a quadratic equation are real or complex is determined by the coefficients of the quadratic equation in standard form (ax2+bx+c=0).
Diverse Learning Needs:
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
12. Select an appropriate method to solve a system of two linear equations in two variables.

a. Solve a system of two equations in two variables by using linear combinations; contrast situations in which use of linear combinations is more efficient with those in which substitution is more efficient.

b. Contrast solutions to a system of two linear equations in two variables produced by algebraic methods with graphical and tabular methods. [Algebra I with Probability, 10]
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Use elimination to solve a system of linear equations.
  • Select an appropriate method (graphical, tabular, substitution, or elimination) for solving a system of linear equations.
  • Decide when substitution is a more efficient method than elimination and vice versa.
  • Contrast solutions produced algebraically with those produced using a table or graph.
  • Provide reasonable approximations when appropriate in a graph or table.
Teacher Vocabulary:
  • Elimination
  • Substitution
  • Graph
  • Table
  • Solution to a system of linear equations
Knowledge:
Students know:
  • Algebraic techniques for manipulating and solving equations.
Skills:
Students are able to:
  • graph a system of linear equations.
  • Generate a table from an equation.
  • Find the solution to a system by graphing, completing a table, substitution, and elimination.
  • Justify which method they used.
Understanding:
Students understand that:
  • When the properties of operations and equality are applied to systems of equations, the resulting equations have the same solution as the original.
Diverse Learning Needs:
Expressions, equations, and inequalities can be used to analyze and make predictions, both within mathematics and as mathematics is applied in different contexts - in particular, contexts that arise in relation to linear, quadratic, and exponential situations.
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
13. Create equations and inequalities in one variable and use them to solve problems in context, either exactly or approximately. Extend from contexts arising from linear functions to those involving quadratic, exponential, and absolute value functions. [Algebra I with Probability, 11]
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a contextual situation that may include linear, quadratic, exponential, or absolute value relationships in one variable,
  • Model the relationship with equations or inequalities and solve the problem presented in the contextual situation for the given variable.
Teacher Vocabulary:
  • Inequality
  • Variable
  • Solution set
  • Linear relationship -Quadratic relationship
  • Exponential relationship
  • Absolute value
Knowledge:
Students know:
  • When the situation presented in a contextual problem is most accurately modeled by a linear, quadratic, exponential, or absolute value relationship.
Skills:
Students are able to:
  • Write equations or inequalities in one variable that accurately model contextual situations.
  • Solve equations and inequalities.
Understanding:
Students understand that:
  • Features of a contextual problem can be used to create a mathematical model for that problem.
Diverse Learning Needs:
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
14. Create equations in two or more variables to represent relationships between quantities in context; graph equations on coordinate axes with labels and scales and use them to make predictions. Limit to contexts arising from linear, quadratic, exponential, absolute value, and linear piecewise functions. [Algebra I with Probability, 12]
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a contextual situation expressing a relationship (linear, quadratic, exponential, absolute value, and linear piecewise functions) between quantities with two or more variables,
  • Model the relationship with equations.
  • Graph the relationship on coordinate axes with labels and scales.
  • Make predictions from the graphs.
Teacher Vocabulary:
  • Linear
  • Quadratic
  • Exponential
  • Absolute Value
  • Linear Piecewise Function
  • x and y axes
Knowledge:
Students know:
  • When the situation presented in a contextual problem is most accurately modeled by a linear, quadratic, exponential, absolute value or linear piecewise function.
  • How to graph equations on a coordinate plane.
Skills:
Students are able to:
  • Write equations with two or more variables.
  • Graph equations on a coordinate plane with appropriate scales and labels.
  • Make predictions based on the graph.
Understanding:
Students understand that:
  • There are relationships among features of a contextual problem, a created mathematical model for that problem, and a graph of that relationship.
  • Models can be used to make predictions.
Diverse Learning Needs:
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
15. Represent constraints by equations and/or inequalities, and solve systems of equations and/or inequalities, interpreting solutions as viable or nonviable options in a modeling context. Limit to contexts arising from linear, quadratic, exponential, absolute value, and linear piecewise functions. [Algebra I with Probability, 13]
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a contextual situation involving constraints,
  • Write equations or inequalities or a system of equations or inequalities that model the situation and justify each part of the model in terms of the context,
  • Solve the equation, inequalities or systems and interpret the solution in the original context including discarding solutions to the mathematical model that cannot fit the real world situation (distance cannot be negative),
  • Solve a system by graphing the system on the same coordinate grid and determine the point(s) or region that satisfies all members of the system,
  • Determine the point(s) of the region satisfying all members of the system that maximizes or minimizes the variable of interest in the case of a system of inequalities.
Note: Contexts can lead to linear, quadratic, exponential, absolute value, and linear piecewise functions.
Teacher Vocabulary:
  • Constraints
  • System of equations
  • System of inequalities
  • Solutions
  • Feasible region
  • viable and non
  • viable options
  • linear
  • quadratic
  • Exponential
  • Absolute value
  • linear piecewise
  • Profit
  • Boundary
  • Closed half plane
  • Open half plane
  • Half plane
  • Consistent
  • Inconsistent
  • Dependent
  • Independent
  • Region
Knowledge:
Students know:
  • When a particular system of two variable equations or inequalities accurately models the situation presented in a contextual problem,
  • Which points in the solution of a system of linear inequalities need to be tested to maximize or minimize the variable of interest.
Skills:
Students are able to:
  • Graph equations and inequalities involving two variables on coordinate axes.
  • Identify the region that satisfies both inequalities in a system.
  • Identify the point(s) that maximizes or minimizes the variable of interest in a system of inequalities.
  • Test a mathematical model using equations, inequalities, or a system against the constraints in the context and interpret the solution in this context.
Understanding:
Students understand that:
  • A symbolic representation of relevant features of a real world problem can provide for resolution of the problem and interpretation of the situation and solution.
  • Representing a physical situation with a mathematical model requires consideration of the accuracy and limitations of the model.
Diverse Learning Needs:
Functions shift the emphasis from a point-by-point relationship between two variables (input/output) to considering an entire set of ordered pairs (where each first element is paired with exactly one second element) as an entity with its own features and characteristics.
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
16. Define a function as a mapping from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range. [Grade 8, 13, edited for added content]

a. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. [Grade 8, 14, edited for added content]
Note: If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x.

b. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Limit to linear, quadratic, exponential, and absolute value functions. [Algebra I with Probability, 15]
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a functional relationship,
  • Determine that exactly one element of the range (output) is assigned to each element of the domain (input) by the function,
  • Represent the function with a graph and with functional notation.
  • Evaluate functional notation to produce a range when given a value in the domain,
  • Explain in the original context the meaning of the output when related to the input.
Given a contextual situation that is functional,
  • Model the situation with a graph and construct the graph based on the parameters given in the domain of the context.
  • Distinguish between those that are functions and non-functions.
Teacher Vocabulary:
  • Function
  • Relation
  • Mapping
  • Domain
  • Range
  • Functional notation f(x)
  • Element
  • Input
  • output
  • Quantitative relationship
Knowledge:
Students know:
  • Distinguishing characteristics of functions,
  • Conventions of function notation,
  • Techniques for graphing functions,
  • Techniques for determining the domain of a function from its context.
Skills:
Students are able to:
  • Accurately graph functions when given function notation.
  • Accurately evaluate function equations given values in the domain.
  • Interpret the domain from the context,
  • Produce a graph of a function based on the context given.
Understanding:
Students understand that:
  • Functions are relationships between two variables that have a unique characteristic: that for each input there exists exactly one output.
  • Function notation is useful to see the relationship between two variables when the unique output for each input relation is satisfied.
  • Different contexts produce different domains and graphs.
  • Function notation in itself may produce graph points which should not be in the graph as the domain is limited by the context.
Diverse Learning Needs:
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
17. Given a relation defined by an equation in two variables, identify the graph of the relation as the set of all its solutions plotted in the coordinate plane. [Algebra I with Probability, 14]
Note: The graph of a relation often forms a curve (which could be a line).
Unpacked Content
Evidence Of Student Attainment:
Students:
Given an equation in two variables,
  • Verify that any ordered pair that makes the equation true is a point on the graph,
  • Show that there are an infinite number of ordered pairs that satisfy the equation.
Teacher Vocabulary:
  • Relation
  • Curve (which could be a line)
  • Graphically Finite solutions
  • Infinite solutions
Knowledge:
Students know:
  • Appropriate methods to find ordered pairs that satisfy an equation,
  • Techniques to graph the collection of ordered pairs to form a curve.
Skills:
Students are able to:
  • Accurately find ordered pairs that satisfy the equation.
  • Accurately graph the ordered pairs and form a curve.
Understanding:
Students understand that:
  • An equation in two variables has an infinite number of solutions (ordered pairs that make the equation true), and those solutions can be represented by the graph of a curve.
Diverse Learning Needs:
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
18. Compare and contrast relations and functions represented by equations, graphs, or tables that show related values; determine whether a relation is a function. Identify that a function f is a special kind of relation defined by the equation y = f(x). [Algebra I with Probability, 16]
Unpacked Content
Evidence Of Student Attainment:
Students:
Given relations between two variables in graphical form, set of ordered pairs, tables, mappings, or equations,
  • Distinguish between those that are functions and non-functions.
  • Use key features to compare and contrast the relations.
  • Explain and justify the similarities and differences of the relations.
Teacher Vocabulary:
  • Function
  • Relation
  • vertical line test
Knowledge:
Students know:
  • In graphing functions the ordered pairs are (x,f(x)) and the graph is y = f(x).
  • Techniques for graphing functions.
  • Techniques to find key features of functions when presented in different ways.
  • Techniques to convert a function to a different form (algebraically, graphically, numerically in tables, or by verbal descriptions).
  • The vertical line test can be used to determine if a graph is a function.
  • A function is a special kind of relation.
Skills:
Students are able to:
  • Accurately determine which key features are most appropriate for comparing functions.
  • Manipulate functions algebraically to reveal key functions.
  • Convert a function to a different form (algebraically, graphically, numerically in tables, or by verbal descriptions) for the purpose of comparing it to another function.
Understanding:
Students understand that:
  • Functions can be written in different but equivalent ways (algebraically, graphically, numerically in tables, or by verbal descriptions).
  • Different representations of functions may aid in comparing key features of the functions.
Diverse Learning Needs:
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a contextual situation containing two quantities,
  • Create a new function and evaluate it by using standard function types and arithmetic operations to combine the original functions to model the relationship of the given quantities.
  • Interpret the function composition in context.
Teacher Vocabulary:
  • Function composition
Knowledge:
Students know:
  • Techniques for expressing functional relationships between two quantities.
  • Techniques to combine functions using arithmetic operations.
Skills:
Students are able to:
  • Accurately develop a model that shows the functional relationship between two quantities.
  • Accurately create a new function through arithmetic operations of other functions.
  • Present an argument to show how the function models the relationship between the quantities.
Understanding:
Students understand that:
  • Relationships can be modeled by several methods.
  • Arithmetic combinations of functions may be used to improve the fit of a model.
Diverse Learning Needs:
Graphs can be used to obtain exact or approximate solutions of equations, inequalities, and systems of equations and inequalities - including systems of linear equations in two variables and systems of linear and quadratic equations (given or obtained by using technology).
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
20. 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).

a. Find the approximate solutions of an equation graphically, using tables of values, or finding successive approximations, using technology where appropriate. [Algebra I with Probability, 19]
Note: Include cases where f(x) is linear, quadratic, exponential, or absolute value functions and g(x) is constant or linear.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given two functions where one is linear, quadratic, exponential, or absolute value functions and the other is constant or linear,
  • Graph each function and identify the intersection point(s).
  • Explain solutions for f(x) = g(x) as the x-coordinate of the points of intersection of the graphs, and explain solution paths.
  • Use technology, tables, and successive approximations to produce the graphs, as well as to determine the approximation of solutions.
Teacher Vocabulary:
  • Functions
  • Successive approximations
  • Linear functions
  • Quadratic functions
  • Absolute value functions
  • Exponential functions
  • Intersection point(s)
Knowledge:
Students know:
  • Defining characteristics of linear, quadratic, absolute value, and exponential graphs.
  • Methods to use technology, tables, and successive approximations to produce graphs and tables.
Skills:
Students are able to:
  • Determine a solution or solutions of a system of two functions.
  • Accurately use technology to produce graphs and tables for linear, quadratic, absolute value, and exponential functions.
  • Accurately use technology to approximate solutions on graphs.
Understanding:
Students understand that:
  • When two functions are equal, the x coordinate(s) of the intersection of those functions is the value that produces the same output (y-value) for both functions.
  • Technology is useful to quickly and accurately determine solutions and produce graphs of functions.
Diverse Learning Needs:
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
21. Graph the solutions to 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 to a system of linear inequalities in two variables as the intersection of the corresponding half-planes, using technology where appropriate. [Algebra I with Probability, 20]
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Given a linear inequality in two variables or a system of linear inequalities, graph solutions and solution sets using the appropriate notation (dotted or solid line) and shadings.
Teacher Vocabulary:
  • Half-planes
  • System of linear inequalities
  • Boundaries
  • Closed half-plane
  • Open half-plane
Knowledge:
Students know:
  • When to include and exclude the boundary of linear inequalities.
  • Techniques to graph the boundaries of linear inequalities.
  • Methods to find solution regions of a linear inequality and systems of linear inequalities.
Skills:
Students are able to:
  • Accurately graph a linear inequality and identify values that make the inequality true (solutions).
  • Find the intersection of multiple linear inequalities to solve a system.
  • Use technology to graph inequalities and systems of inequalities.
Understanding:
Students understand that:
  • Solutions to a linear inequality result in the graph of a half-plane.
  • Solutions to a system of linear inequalities are the intersection of the solutions of each inequality in the system.
Diverse Learning Needs:
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
22. Solve systems consisting of linear and/or quadratic equations in two variables graphically, using technology where appropriate. [Algebra I with Probability, 18]
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a system of a linear equations and/or a quadratic equations,
  • Graph the linear equation and the quadratic equation on the same Cartesian plane, and identify the intersection point(s).
  • Make sense of the existence of 0, 1, or 2 solutions to the system by explaining the relationship of the solutions to the graph.
  • Verify that the proposed solutions satisfy both equations.
Teacher Vocabulary:
  • Solving systems of equations
  • System of equations
  • Cartesian plane
  • Substitution
Knowledge:
Students know:
  • The conditions under which a linear equation and a quadratic equation have 0, 1, or 2 solutions.
  • Techniques for producing and interpreting graphs of linear and quadratic equations.
  • Appropriate use of properties of equality.
Skills:
Students are able to:
  • Graph linear and quadratic equations precisely and interpret the results.
  • Use technology to graph systems of equations.
Understanding:
Students understand that:
  • Solutions of a system of equations is the set of all ordered pairs that make both equations true simultaneously.
  • A system consisting of a linear equation and a quadratic equation will have 0, 1, or 2 solutions.
Diverse Learning Needs:
Functions can be described by using a variety of representations: mapping diagrams, function notation (e.g., f(x) = x2), recursive definitions, tables, and graphs.
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
23. Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Include linear, quadratic, exponential, absolute value, and linear piecewise. [Algebra I with Probability, 21, edited]

a. Distinguish between linear and non-linear functions. [Grade 8, 15a]
Unpacked Content
Evidence Of Student Attainment:
Students:
Given functions represented in various ways(algebraically, graphically, numerically in tables, or by verbal descriptions).
  • Use key features to compare the functions.
  • Explain and justify the similarities and differences of the functions.

  • Given a variety of functions in equation form, use logical reasoning to justify their classification as linear or non-linear by interpreting the relationships in the expressions.
Teacher Vocabulary:
  • Linear function
  • Exponential function
  • Quadratic function
  • Absolute value function
  • Linear Piecewise function
  • non-linear functions
Knowledge:
Students know:
  • Techniques to find key features of functions when presented in different ways.
  • Techniques to convert a function to a different form (algebraically, graphically, numerically in tables, or by verbal descriptions).
  • Characteristics of linear and nonlinear functions.
Skills:
Students are able to:
  • Accurately determine which key features are most appropriate for comparing functions.
  • Manipulate functions algebraically to reveal key functions.
  • Convert a function to a different form (algebraically, graphically, numerically in tables, or by verbal descriptions) for the purpose of comparing it to another function.
  • Compare functions based on their properties.
Understanding:
Students understand that:
  • Functions can be written in different but equivalent ways (algebraically, graphically, numerically in tables, or by verbal descriptions).
  • Different representations of functions may aid in comparing key features of the functions.
  • Functions are relationships between two variables that have a unique characteristic, that being, for each input there exists exactly one output.
  • Functions can be represented in a variety of ways (graphs, tables, and equations), each of which provides unique perspectives of the relationship between the variables.
  • Linear functions have a defining characteristic of a unit rate or slope that other nonlinear functions do not have.
Diverse Learning Needs:
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
24. Define sequences as functions, including recursive definitions, whose domain is a subset of the integers.

a. Write explicit and recursive formulas for arithmetic and geometric sequences and connect them to linear and exponential functions. [Algebra I with Probability, 22]

Example: A sequence with constant growth will be a linear function, while a sequence with proportional growth will be an exponential function.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a contextual situation that is sequential (arithmetic and geometric),
  • Create both recursive and explicit models for the sequence.
  • Explain and justify the relationship between the recursive and explicit forms that model the situation.
  • Generate and justify a function that relates the number of the term to the value of the term in the sequence.
Teacher Vocabulary:
  • Sequence
  • Recursively
  • Domain
  • Arithmetic sequence
  • Geometric sequence
Knowledge:
Students know:
  • Distinguishing characteristics of a function.
  • Distinguishing characteristics of function notation.
  • Distinguishing characteristics of generating sequences.
Skills:
Students are able to:
  • Use the properties of operations and equality and knowledge of recursive functions to justify that an explicit formula that models a sequence is equivalent to a recursive model.
Understanding:
Students understand that:
  • Each term in the domain of a sequence defined as a function is unique and consecutive.
Diverse Learning Needs:
Functions that are members of the same family have distinguishing attributes (structure) common to all functions within that family
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
25. Identify the effect on the graph of replacing f(x) by f(x) + k, k ยท 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 explain the effects on the graph, using technology as appropriate. Extend from linear to quadratic, exponential, absolute value, and linear piecewise functions. [Algebra I with Probability, 23, edited]
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a function in algebraic form,
  • Graph the function, f(x), conjecture how the graph of f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k(both positive and negative) will change from f(x), and test the conjectures,
  • Describe how the graphs of the functions were affected (e.g., horizontal and vertical shifts, horizontal and vertical stretches, or reflections),
  • Use technology to explain possible effects on the graph from adding or multiplying the input or output of a function by a constant value,
  • Recognize if a function is even or odd.


Given the graph of a function and the graph of a translation, stretch, or reflection of that function,
  • Determine the value which was used to shift, stretch, or reflect the graph.
  • Recognize if a function is even or odd.
Teacher Vocabulary:
  • Even and odd functions
  • Composite functions
  • Horizontal and vertical shifts
  • Horizontal and vertical stretch
  • Reflections
  • Translations
Knowledge:
Students know:
  • Graphing techniques of functions,
  • Methods of using technology to graph functions.
  • Techniques to identify even and odd functions both algebraically and from a graph.
Skills:
Students are able to:
  • Accurately graph functions.
  • Check conjectures about how a parameter change in a function changes the graph and critique the reasoning of others about such shifts.
  • Identify shifts, stretches, or reflections between graphs.
  • Determine when a function is even or odd.
Understanding:
Students understand that:
  • Graphs of functions may be shifted, stretched, or reflected by adding or multiplying the input or output of a function by a constant value.
  • Even and odd functions may be identified from a graph or algebraic form of a function.
Diverse Learning Needs:
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
26. Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.

a. Show that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.

b. Define linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Define exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. [Algebra I with Probability, 24]
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a linear or exponential function,
  • Create a sequence from the functions and examine the results to demonstrate that linear functions grow by equal differences, and exponential functions grow by equal factors over equal intervals.
  • Use slope-intercept form of a linear function and the general definition of exponential functions to justify through algebraic rearrangements that linear functions grow by equal differences, and exponential functions grow by equal factors over equal intervals.

  • Given a contextual situation modeled by functions, determine if the change in the output per unit interval is a constant being added or multiplied to a previous output, and appropriately label the function as linear, exponential, or neither.
Teacher Vocabulary:
  • Linear functions
  • Exponential functions
  • Constant rate of change
  • Constant percent rate of change
  • Intervals
  • Percentage of growth
  • Percentage of decay
  • Slope-intercept form of a line
Knowledge:
Students know:
  • Key components of linear and exponential functions.
  • Properties of operations and equality
Skills:
Students are able to:
  • Accurately determine relationships of data from a contextual situation to determine if the situation is one in which one quantity changes at a constant rate per unit interval relative to another (linear).
  • Accurately determine relationships of data from a contextual situation to determine if the situation is one in which one quantity grows or decays by a constant percent rate per unit interval relative to another (exponential).
Understanding:
Students understand that:
  • Linear functions have a constant value added per unit interval, and exponential functions have a constant value multiplied per unit interval.
  • Distinguishing key features of and categorizing functions facilitates mathematical modeling and aids in problem resolution.
Diverse Learning Needs:
Mathematics (2019)
Grade(s): 8
Accelerated
All Resources: 0
27. 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). [Algebra I with Probability, 25]
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a contextual situation shown by a graph, a description of a relationship, or two input-output pairs,
  • Create a linear or exponential function that models the situation.
  • Create arithmetic and geometric sequences from the given situation.
  • Justify the equality of the sequences and the functions mathematically and in terms of the original sequence.
Teacher Vocabulary:
  • Arithmetic sequence
  • Geometric sequence
  • Linear function
  • Exponential function
  • Knowledge:
    Students know:
    • That linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
    • Properties of arithmetic and geometric sequences.
    Skills:
    Students are able to:
    • Accurately recognize relationships within data and use that relationship to create a linear or exponential function to model the data of a contextual situation.
    Understanding:
    Students understand that:
    • Linear and exponential functions may be used to model data that is presented as a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
    • Linear functions have a constant value added per unit interval, and exponential functions have a constant value multiplied per unit interval.
    Diverse Learning Needs:
    Mathematics (2019)
    Grade(s): 8
    Accelerated
    All Resources: 0
    28. Use graphs and tables to show that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically. [Algebra I with Probability, 26]
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given a quantity increasing exponentially and a quantity increasing linearly or quadratically.
    • Construct graphs and tables that demonstrate the exponential function will exceed the linear or quadratic function at some point.
    • Present a convincing argument that this must be true for all polynomial functions.
    Teacher Vocabulary:
    • Increasing exponentially
    • Increasing quadratically
    • Increasing linearly
    Knowledge:
    Students know:
    • Techniques to graph and create tables for exponential, linear, and quadratic functions.
    Skills:
    Students are able to:
    • Accurately create graphs and tables for exponential, linear, and quadratic functions.
    • Use the graphs and tables to present a convincing argument that the exponential function eventually exceeds the linear and quadratic function.
    Understanding:
    Students understand that:
    • Exponential functions grow at a faster rate than linear and quadratic functions after some point in their domain.
    Diverse Learning Needs:
    Mathematics (2019)
    Grade(s): 8
    Accelerated
    All Resources: 0
    29. Interpret the parameters of functions in terms of a context. Extend from linear functions, written in the form mx + b, to exponential functions, written in the form abx. [Algebra I with Probability, 27]

    Example: If the function V(t) = 19885(0.75)t describes the value of a car after it has been owned for t years, 19885 represents the purchase price of the car when t = 0, and 0.75 represents the annual rate at which its value decreases.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given a contextual situation,
    • Create a function that models the situation as linear in the form (mx+b) or exponential in the form (abx).
    • Define and justify the parameters (all constants used to define the function) in terms of the original context.
    Teacher Vocabulary:
    • Parameters
    Knowledge:
    Students know:
    • Key components of linear and exponential functions.
    Skills:
    Students are able to:
    • Communicate the meaning of defining values (parameters and variables) in functions used to model contextual situations in terms of the original context.
    Understanding:
    Students understand that:
    • Sense making in mathematics requires that meaning is attached to every value in a mathematical expression.
    Diverse Learning Needs:
    Functions can be represented graphically and key features of the graphs, including zeros, intercepts, and, when relevant, rate of change and maximum/minimum values, can be associated with and interpreted in terms of the equivalent symbolic representation.
    Mathematics (2019)
    Grade(s): 8
    Accelerated
    All Resources: 0
    30. 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. Note: Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries; and end behavior. Extend from relationships that can be represented by linear functions to quadratic, exponential, absolute value, and general piecewise functions. [Algebra I with Probability, 28]
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given a function that models a relationship between two quantities,
    • Produce the graph and table of the function and show the key features (intercepts. intervals where the function is increasing, decreasing, positive, or negative. relative maximums and minimums. symmetries. and end behavior) that are appropriate for the function.

    • Given key features from verbal description of a relationship, sketch a graph with the given key features.
    Teacher Vocabulary:
    • Function
    • Intercepts
    • Intervals of Increasing
    • Intervals of decreasing
    • Function is positive
    • Function is negative
    • Relative Maximum
    • Relative Minimum
    • Axis symmetry
    • Origin symmetry
    • End behavior
    Knowledge:
    Students know:
    • Key features of function graphs (i.e., intercepts. intervals where the function is increasing, decreasing, positive, or negative. relative maximums and minimums. symmetries. and end behavior).
    • Methods of modeling relationships with a graph or table.
    Skills:
    Students are able to:
    • Accurately graph any relationship.
    • Interpret key features of a graph.
    Understanding:
    Students understand that:
    • The relationship between two variables determines the key features that need to be used when interpreting and producing the graph.
    Diverse Learning Needs:
    Mathematics (2019)
    Grade(s): 8
    Accelerated
    All Resources: 0
    31. 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. Limit to linear, quadratic, exponential, and absolute value functions. [Algebra I with Probability, 29]
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given a function presented as an equation or table,
    • Calculate the average rate of change within a given interval.
    • Interpret the average rate of change for the interval in context.

    • Given a graph of a contextual situation, estimate the rate of change between intervals that are appropriate for the summary of the context.

    Note: Include linear, quadratic, exponential, absolute value, and general piecewise function.
    Teacher Vocabulary:
    • Average rate of change
    • Intervals
    Knowledge:
    Students know:
    • Techniques for graphing.
    • Techniques for finding a rate of change over an interval on a table or from an equation.
    • Techniques for estimating a rate of change over an interval on a graph.
    Skills:
    Students are able to:
    • Calculate rate of change over an interval on a table or from an equation.
    • Estimate a rate of change over an interval on a graph.
    Understanding:
    Students understand that:
    • The average provides information on the overall changes within an interval, not the details within the interval (an average of the endpoints of an interval does not tell you the significant features within the interval).
    Diverse Learning Needs:
    Mathematics (2019)
    Grade(s): 8
    Accelerated
    All Resources: 0
    32. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

    a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

    b. Graph piecewise-defined functions, including step functions and absolute value functions.

    c. Graph exponential functions, showing intercepts and end behavior. [Algebra I with Probability, 30]
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given a symbolic representation of a function (including linear, quadratic, absolute value, piecewise-defined functions, and exponential,
    • Produce an accurate graph (by hand in simple cases and by technology in more complicated cases) and justify that the graph is an alternate representation of the symbolic function.
    • Identify key features of the graph and connect these graphical features to the symbolic function, specifically for special functions:
      A. quadratic or linear (intercepts, maxima, and minima) and piecewise-defined functions, including step functions and absolute value functions (descriptive features such as the values that are in the range of the function and those that are not).
      b. exponential (intercepts and end behavior).
    Teacher Vocabulary:
    • x-intercept
    • y-intercept
    • Maximum
    • Minimum
    • End behavior
    • Linear function
    • Factorization
    • Quadratic function
    • Intercepts
    • Piecewise function
    • Step function
    • Absolute value function
    • Exponential function
    • Domain
    • Range
    • Period
    • Midline
    • Amplitude
    • Zeros
    Knowledge:
    Students know:
    • Techniques for graphing.
    • Key features of graphs of functions.
    Skills:
    Students are able to:
    • Identify the type of function from the symbolic representation.
    • Manipulate expressions to reveal important features for identification in the function.
    • Accurately graph relationships.
    Understanding:
    Students understand that:
    • Key features are different depending on the function.
    • Identifying key features of functions aid in graphing and interpreting the function.
    Diverse Learning Needs:
    Functions model a wide variety of real situations and can help students understand the processes of making and changing assumptions, assigning variables, and finding solutions to contextual problems.
    Mathematics (2019)
    Grade(s): 8
    Accelerated
    All Resources: 0
    33. Use the mathematical modeling cycle to solve real-world problems involving linear, quadratic, exponential, absolute value, and linear piecewise functions. [Algebra I with Probability, 31]
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Engage in the Mathematical Modeling Cycle (Appendix E) to solve contextual problems involving linear, quadratic, exponential, absolute value and linear piecewise function
    Teacher Vocabulary:
    • Mathematical Modeling Cycle
    • Define a problem
    • Make assumptions
    • Define variables
    • Do the math and get solutions
    • Implement and report results
    • Iterate to refine and extend a model
    • Assess a model and solutions
    Knowledge:
    Students know:
    • The Mathematical Modeling Cycle.
    • When to use the Mathematical Modeling Cycle to solve problems.
    Skills:
    Students are able to:
    • Define the problem to be answered.
    • Make assumptions to simplify the problem, identifying the variables in the situation and create an equation.
    • Analyze and perform operations to draw conclusions.
    • Assess the model and solutions in terms of the original context.
    • Refine and extend the model as needed.
    • Report on conclusions and reasonings.
    Understanding:
    Students understand that:
    • Making decisions, evaluating those decisions, and revisiting and revising work is crucial in mathematics and life.
    • Mathematical modeling uses mathematics to answer real-world, complex problems.
    Diverse Learning Needs:
    Mathematics (2019)
    Grade(s): 8
    Accelerated
    All Resources: 0
    34. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities, describing patterns in terms of positive, negative, or no association, linear and non-linear association, clustering, and outliers. [Grade 8, 18]
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given sets of bivariate measurement data or contextual situations in which bivariate measurement data must be collected,
    • Construct and interpret scatter plots,
    • Describe visual patterns observed, (e.g., clustering, outliers, positive or negative association, linear, and nonlinear association).
    Teacher Vocabulary:
    • Scatter plots
    • Bivariate measurement data
    • Clustering
    • Outliers
    • Positive and negative association
    • No association
    • Linear and nonlinear association
    Knowledge:
    Students know:
    • Representations for bivariate data and techniques for constructing each (e.g., tables, scatter plots).
    Skills:
    Students are able to:
    • Construct a scatter plot to represent a set of bivariate data.
    • Use mathematical vocabulary to describe and interpret patterns in bivariate data.
    Understanding:
    Students understand that:
    • Using different representations and descriptors of a data set can be useful in seeing important features of the situation being investigated.
    • Negative association in bivariate data can be a very strong association but is an inverse relationship.
    Diverse Learning Needs:
    Mathematics (2019)
    Grade(s): 8
    Accelerated
    All Resources: 0
    35. Given a scatter plot that suggests a linear association, informally draw a line to fit the data, and assess the model fit by judging the closeness of the data points to the line. [Grade 8, 19]
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given a variety of scatterplots representing bivariate data,
    • Determine if the plots suggest a linear relationship.
    • Informally fit a straight line to the data.
    • Assess the model fit by judging the closeness of the data points to the line.
    Teacher Vocabulary:
    • Scatter plot
    • Linear association
    • Quantitative variable
    Knowledge:
    Students know:
    • Patterns found on scatter plots of bivariate data, (e.g., linear/non-linear, positive/negative).
    • Strategies for informally fitting straight lines to bivariate data with a linear relationship.
    • Methods for finding the distance between two points on a coordinate plane and between a point and a line.
    Skills:
    Students are able to:
    • Construct a scatter plot to represent a set of bivariate data.
    • Use mathematical vocabulary to describe and interpret patterns in bivariate data.
    • Use logical reasoning and appropriate strategies to draw a straight line to fit data that suggest a linear association.
    • Use mathematical vocabulary, logical reasoning, and closeness of data points to a line to judge the fit of the line to the data.
    Understanding:
    Students understand that:
    • Using different representations and descriptors of a data set can be useful in seeing important features of the situation being investigated.
    • When visual examination of a scatter plot suggests a linear association in the data, fitting a straight line to the data can aid in interpretation and prediction.
    Diverse Learning Needs:
    Mathematics (2019)
    Grade(s): 8
    Accelerated
    All Resources: 0
    36. Use a linear model of a real-world situation to solve problems and make predictions.

    a. Describe the rate of change and y-intercept in the context of a problem using a linear model of a real-world situation. [Grade 8, 20]
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given a contextual or mathematical situation involving bivariate measurement data,
    • Represent the situation graphically and algebraically, describe the relationship between the two models, and interpret the slope and the y-intercept of the line in order to find answers to questions.
    • Use the equation and graph to make predictions about unobserved data in context.
    Teacher Vocabulary:
    • Linear model
    • Bivariate measurement data
    • Slope
    • y-intercept
    Knowledge:
    Students know:
    • Strategies for determining slope and y-intercept of a linear model.
    Skills:
    Students are able to:
    • Represent contextual and mathematical situations involving bivariate measurement data with a linear relationship algebraically and graphically.
    • Use mathematical vocabulary to describe and interpret slopes and y-intercepts of lines which represent contextual situations involving bivariate data.
    • Make predictions about unobserved data using the equation and graph.
    Understanding:
    Students understand that:
    • Modeling bivariate data with scatter plots and fitting a straight line to the data can aid in interpretation of the data and predictions about unobserved data.
    Diverse Learning Needs:
    Mathematics (2019)
    Grade(s): 8
    Accelerated
    All Resources: 0
    37. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects, using relative frequencies calculated for rows or columns to describe possible associations between the two variables. [Grade 8, 21]
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given a contextual or mathematical situation involving bivariate categorical data,
    • Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects.
    • Use relative frequencies calculated for rows or columns to describe possible association between the two variables.
    Teacher Vocabulary:
    • Two-way table
    • Rows
    • Columns
    • Bivariate categorical data
    • Frequencies
    • Relative frequencies
    • Categorical variables
    Knowledge:
    Students know:
    • Characteristics of data sets that distinguish categorical data from measurement data.
    Skills:
    Students are able to:
    • Construct two-way tables for categorical data.
    • Find relative frequencies for cells in the two-way tables.
    • Conjecture about patterns of association in the two-way tables and explain the reasoning that leads to the conjecture.
    Understanding:
    Students understand that:
    • Organizing categorical data in two-way tables can aid in identifying patterns of association in the data.
    • Relative frequencies, rather than just absolute frequencies, need to be calculated from two-way tables to identify patterns of association.
    Diverse Learning Needs:
    Data arise from a context and come in two types: quantitative (continuous or discrete) and categorical. Technology can be used to "clean" and organize data, including very large data sets, into a useful and manageable structure - a first step in any analysis of data.
    Mathematics (2019)
    Grade(s): 8
    Accelerated
    All Resources: 0
    38. Distinguish between quantitative and categorical data and between the techniques that may be used for analyzing data of these two types. [Algebra I with Probability, 34]

    Example: The color of cars is categorical and so is summarized by frequency and proportion for each color category, while the mileage on each car's odometer is quantitative and can be summarized by the mean.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Compare quantitative and categorical data.
    • Compare the techniques that may be used to analyze quantitative and categorical data
    Teacher Vocabulary:
    • Quantitative data
    • Categorical data
    • Mean
    • Median
    • Mode
    • Frequency
    Knowledge:
    Students know:
    • Characteristics of quantitative data.
    • Characteristics of categorical data.
    • Techniques for analyzing categorical data.
    • Techniques for analyzing quantitative data.
    Skills:
    Students are able to:
    • Organize quantitative (continuous or discrete) data in different ways.
    • Organize categorical data in different ways.
    • Analyze data in meaningful ways.
    Understanding:
    Students understand that:
    • Quantitative data can be analyzed for example using measures of center and variability.
    • Categorical data can be analyzed for example using frequency and proportion.
    Diverse Learning Needs:
    The association between two categorical variables is typically represented by using two-way tables and segmented bar graphs.
    Mathematics (2019)
    Grade(s): 8
    Accelerated
    All Resources: 0
    39. Analyze the possible association between two categorical variables.

    a. Summarize categorical data for two categories in two-way frequency tables and represent using segmented bar graphs.

    b. Interpret relative frequencies in the context of categorical data (including joint, marginal, and conditional relative frequencies).

    c. Identify possible associations and trends in categorical data. [Algebra I with Probability, 35]
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given categorical data for two categories,
    • Create two-way frequency tables.
    • Represent the data using segmented bar graphs.
    • Find relative frequencies using ratios.
    • Recognize and justify possible relationships and patterns in the data by examining the joint, marginal, and conditional relative frequencies.
    • Identify possible associations and trends in the data.
    Teacher Vocabulary:
    • Categorical data
    • two-Way frequency Tables
    • Relative frequency
    • Joint frequency
    • Marginal frequency
    • Conditional relative frequency
    Knowledge:
    Students know:
    • Characteristics of a two-way frequency table.
    • Methods for converting frequency tables to relative frequency tables.
    • That the sum of the frequencies in a row or a column gives the marginal frequency.
    • Techniques for finding conditional relative frequency.
    • Techniques for finding the joint frequency in tables.
    • Use mathematical vocabulary to describe associations and trends in data.
    Skills:
    Students are able to:
    • Accurately construct frequency tables.
    • Accurately construct relative frequency tables.
    • Accurately find the joint, marginal, and conditional relative frequencies.
    • Recognize and explain possible associations and trends in the data.
    Understanding:
    Students understand that:
    • Two-way frequency tables may be used to represent categorical data.
    • Relative frequency tables show the ratios of the categorical data in terms of joint, marginal, and conditional relative frequencies.
    • Two-way frequency or relative frequency tables may be used to aid in recognizing associations and trends in the data.
    Diverse Learning Needs:
    Data analysis techniques can be used to develop models of contextual situations and to generate and evaluate possible solutions to real problems involving those contexts.
    Mathematics (2019)
    Grade(s): 8
    Accelerated
    All Resources: 0
    40. Generate a two-way categorical table in order to find and evaluate solutions to real-world problems.

    a. Aggregate data from several groups to find an overall association between two categorical variables.

    b. Recognize and explore situations where the association between two categorical variables is reversed when a third variable is considered (Simpson's Paradox). [Algebra I with Probability, 36]

    Example: In a certain city, Hospital 1 has a higher fatality rate than Hospital 2. But when considering mildly-injured patients and severely-injured patients as separate groups, Hospital 1 has a lower fatality rate among both groups than Hospital 2, since Hospital 1 is a Level 1 Trauma Center. Thus, Hospital 1 receives most of the severely-injured patients who are less likely to survive overall but have a better chance of surviving in Hospital 1 than they would in Hospital 2.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given a situation in which it is meaningful to collect categorical data for many categories.
    • Collect data and create two-way frequency tables.
    • Find and evaluate solutions to real world problems.
    • Sift through and analyze data to find an overall association between two categorical variables.
    • Explore situations of the Simpson's Paradox.
    Teacher Vocabulary:
    • Categorical data
    • Categorical variable
    • two-Way frequency tables
    • Simpson's Paradox
    Knowledge:
    Students know:
    • Techniques to construct two-way frequency tables
    Skills:
    Students are able to:
    • Accurately construct a two-way frequency table.
    • Recognize situations that model Simpson's Paradox.
    Understanding:
    Students understand that:
    • Data analysis techniques can be used to develop models of contextual situations and generate and evaluate possible solutions to real world problems.
    Diverse Learning Needs:
    Mathematical and statistical reasoning about data can be used to evaluate conclusions and assess risks.
    Mathematics (2019)
    Grade(s): 8
    Accelerated
    All Resources: 0
    41. Use mathematical and statistical reasoning with bivariate categorical data in order to draw conclusions and assess risk. [Algebra I with Probability, 32]

    Example: In a clinical trial comparing the effectiveness of flu shots A and B, 21 subjects in treatment group A avoided getting the flu while 29 contracted it. In group B, 12 avoided the flu while 13 contracted it. Discuss which flu shot appears to be more effective in reducing the chances of contracting the flu.
    Possible answer: Even though more people in group A avoided the flu than in group B, the proportion of people avoiding the flu in group B is greater than the proportion in group A, which suggests that treatment B may be more effective in lowering the risk of getting the flu.

    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    By using mathematical and statistical reasoning,
    • Draw conclusions about bivariate categorical data.
    • Assess risk related to bivariate categorical data.
    Teacher Vocabulary:
    • Bivariate categorical data
    • Conclusions
    • Risks
    • Line of best fit
    • Association
    • Trend
    • qualitative literacy
    • Categorical data
    Knowledge:
    Students know:
    • Techniques to construct two-way frequency tables.
    • Key features of bivariate categorical data.
    • Strategies for drawing conclusions.
    • Strategies for assessing risk.
    Skills:
    Students are able to:
    • Accurately construct a two-way frequency table.
    • Draw conclusions from the data.
    • Assess risk from the data.
    Understanding:
    Students understand that:
    • Mathematical and statistical reasoning about data can be used to evaluate conclusions and assess risks.
    Diverse Learning Needs:
    Making and defending informed, data-based decisions is a characteristic of a quantitatively literate person.
    Mathematics (2019)
    Grade(s): 8
    Accelerated
    All Resources: 0
    42. Design and carry out an investigation to determine whether there appears to be an association between two categorical variables, and write a persuasive argument based on the results of the investigation. [Algebra I with Probability, 33]

    Example: Investigate whether there appears to be an association between successfully completing a task in a given length of time and listening to music while attempting to complete the task. Randomly assign some students to listen to music while attempting to complete the task and others to complete the task without listening to music. Discuss whether students should listen to music while studying, based on that analysis.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Design and carry out an investigation to determine what association, if any, exists between two categorical variables.
    • Write a persuasive argument based on the results of the investigation.
    Teacher Vocabulary:
    • Investigation
    • Categorical variables
    • Persuasive argument
    Knowledge:
    Students know:
    • Techniques for collecting and analyzing data.
    • Mathematical vocabulary related to associations (positive, negative, no, linear, non-linear)
    Skills:
    Students are able to:
    • Accurately display data.
    • Accurately analyze the data for associations.
    • Write persuasive arguments based on the data analysis.
    Understanding:
    Students understand that:
    • Making and defending informed, data-based decisions is a characteristic of a quantitatively literate person.
    Diverse Learning Needs:
    Two events are independent if the occurrence of one event does not affect the probability of the other event. Determining whether two events are independent can be used for finding and understanding probabilities.
    Mathematics (2019)
    Grade(s): 8
    Accelerated
    All Resources: 0
    43. 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"). [Algebra I with Probability, 37]
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given scenarios involving chance,
    • Determine the sample space and a variety of simple and compound events that may be defined from the sample space.
    • Use the language of union, intersection, and complement appropriately to define events.
    Teacher Vocabulary:
    • Subsets
    • Sample space
    • Unions
    • Intersections
    • Complements
    • Event
    • Outcome
    Knowledge:
    Students know:
    • Methods for describing events from a sample space using set language (subset, union, intersection, complement).
    Skills:
    Students are able to:
    • Interpret the given information in the problem.
    • Accurately determine the probability of the scenario.
    Understanding:
    Students understand that:
    • Set language can be useful to define events in a probability situation and to symbolize relationships of events.
    Diverse Learning Needs:
    Mathematics (2019)
    Grade(s): 8
    Accelerated
    All Resources: 0
    44. Explain whether two events, A and B, are independent, using two-way tables or tree diagrams. [Algebra I with Probability, 38]
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given scenarios involving two events,
    • Explain the meaning of independence from a formula perspective P(A & B) = P(A) x P(B) and from the intuitive notion that A occurring has no impact on whether B occurs or not.
    • Explain the meaning of independence using a two-way table and tree diagrams.
    • Compare these two interpretations within the context of the scenario.
    Teacher Vocabulary:
    • Independent event
    • Probability
    • Dependent event
    • Event
    • Two-way table
    • Tree diagram
    • Simple event
    • Compound event
    Knowledge:
    Students know:
    • Methods to find probability of simple and compound events.
    Skills:
    Students are able to:
    • Interpret the given information in the problem.
    • Accurately determine the probability of simple and compound events.
    • Accurately calculate the product of the probabilities of two events.
    Understanding:
    Students understand that:
    • Events are independent if one occurring does not affect the probability of the other occurring, and that this may be demonstrated mathematically by showing the truth of P(A & B) = P(A) x P(B).
    Diverse Learning Needs:
    Conditional probabilities = that is, those probabilities that are "conditioned" by some known information = can be computed from data organized in contingency tables. Conditions or assumptions may affect the computation of a probability.
    Mathematics (2019)
    Grade(s): 8
    Accelerated
    All Resources: 0
    45. Compute the conditional probability of event A given event B, using two-way tables or tree diagrams. [Algebra I with Probability, 39]
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given scenarios involving two events A and B both when A and B are independent and when A and B are dependent,
    • Determine the probability of each individual event, then limit the sample space to those outcomes where B has occurred and calculate the probability of A, compare the P(A) and the P(A given B), and explain the equality or difference in the original context of the problem.
    • Justify that P(A given B) = P(A&B)/P(B).
    Teacher Vocabulary:
    • Conditional probability
    • Independence
    • Probability
    • Sample space
    • Simple event
    • Compound event
    Knowledge:
    Students know:
    • Methods to find probability of simple and compound events.
    • Techniques to find conditional probability.
    Skills:
    Students are able to:
    • Accurately determine the probability of simple and compound events.
    • Accurately determine the conditional probability P(A given B) from a sample space or from the knowledge of P(A&B) and the P(B).
    Understanding:
    Students understand that:
    • The independence of two events is determined by the effect that one event has on the outcome of another event.
    • The occurrence of one event may or may not influence the likelihood that another event occurs.
    Diverse Learning Needs:
    Conditional probabilities - that is, those probabilities that are "conditioned" by some known information - can be computed from data organized in contingency tables. Conditions or assumptions may affect the computation of a probability.
    Mathematics (2019)
    Grade(s): 8
    Accelerated
    All Resources: 0
    46. Recognize and describe the concepts of conditional probability and independence in everyday situations and explain them using everyday language. [Algebra I with Probability, 40]

    Example: Contrast the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given a contextual situation and scenarios involving two events,
    • Explain the meaning of independence from a formula perspective P(A & B) = P(A) x P(B) and from the intuitive notion that A occurring has no impact on whether B occurs or not.
    • Compare these two interpretations within the context of the scenario.
    Teacher Vocabulary:
    • Conditional probability
    • Independence
    • Probability
    Knowledge:
    Students know:
    • Possible relationships and differences between the simple probability of an event and the probability of an event under a condition.
    Skills:
    Students are able to:
    • Communicate the concepts of conditional probability and independence using everyday language by discussing the impact of the occurrence of one event on the likelihood of the other occurring.
    Understanding:
    Students understand that:
    • The occurrence of one event may or may not influence the likelihood that another event occurs (e.g., successive flips of a coin
    • First toss exerts no influence on whether a head occurs on the second, drawing an ace from a deck changes the probability that the next card drawn is an ace).
    • Events are independent if the occurrence of one does not affect the probability of the other occurring.
    Diverse Learning Needs:
    Conditional probabilities = that is, those probabilities that are "conditioned" by some known information = can be computed from data organized in contingency tables. Conditions or assumptions may affect the computation of a probability.
    Mathematics (2019)
    Grade(s): 8
    Accelerated
    All Resources: 0
    47. Explain why the conditional probability of A given B is the fraction of B's outcomes that also belong to A, and interpret the answer in context. [Algebra I with Probability, 41]

    Example: the probability of drawing a king from a deck of cards, given that it is a face card, is (4/52)/(12/52), which is 1/3.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given a contextual situation consisting of two events,
    • Determine the probability of each individual event, then limit the sample space to those outcomes where B has occurred and calculate the probability of A, compare the P(A)and the P(A given B), and explain the equality or difference in the original context of the problem.
    • Determine the probability of each individual event, then limit the sample space to those outcomes where B has occurred and calculate the P(A and B), compare the ratio of P(A and B) and P(B) to P(A given B), and explain the equality or difference in the original context of the problem.
    Teacher Vocabulary:
    • Conditional probability
    • Probability
    • Simple events
    • Compound events
    • Sample space
    • Independent events
    • Dependent events
    Knowledge:
    Students know:
    • Possible relationships and differences between the simple probability of an event and the probability of an event under a condition.
    Skills:
    Students are able to:
    • Accurately determine the probability of simple and compound events.
    • Accurately determine the conditional probability P(A given B) from a sample space or from the knowledge of P(A&B) and the P(B).
    Understanding:
    Students understand that:
    • Conditional probability is the probability of an event occurring given that another event has occurred.
    Diverse Learning Needs:
    Geometry and Measurement
    Understand and apply the Pythagorean Theorem.
    Mathematics (2019)
    Grade(s): 8
    Accelerated
    All Resources: 0
    48. Informally justify the Pythagorean Theorem and its converse. [Grade 8, 26]
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Use mathematical reasoning and vocabulary to verbally explain the pythagorean theorem and its converse.
    Teacher Vocabulary:
    • Pythagorean Theorem
    • Converse
    Knowledge:
    Students know:
    • The Pythagorean Theorem.
    • Vocabulary related to right triangles (hypotenuse, leg)
    Skills:
    Students are able to:
    • Use mathematical reasoning and vocabulary to verbally explain a proof of the Pythagorean Theorem and its converse.
    Understanding:
    Students understand that:
    • Theorems represent generalizations about geometric relationships that are used to solve problems.
    Diverse Learning Needs:
    Mathematics (2019)
    Grade(s): 8
    Accelerated
    All Resources: 0
    49. Apply the Pythagorean Theorem to find the distance between two points in a coordinate plane. [Grade 8, 27]
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Given real-world and mathematical problems that can be represented on a coordinate plane, apply the Pythagorean Theorem in order to solve problems and justify solutions and solution paths for finding side lengths (distances between points) in right triangles within the problem contexts.
    Teacher Vocabulary:
    • Pythagorean Theorem
    • Right triangle
    • hypotenuse
    • leg
    • Coordinate plane
    • ordered pair
    Knowledge:
    Students know:
    • Pythagorean Theorem.
    • Operations and labeling within a coordinate system.
    Skills:
    Students are able to:
    • Solve equations involving one variable and square root.
    • Represent real-world and mathematical contexts involving right triangles in a variety of formats (e.g., drawings on coordinate planes, equations).
    • Justify solutions and solution paths using conceptual understandings and vocabulary related to the Pythagorean Theorem (right angle, hypotenuse).
    Understanding:
    Students understand that:
    • The properties of right triangles can be used to solve problems,
    • Theorems represent general relationships that are true for all shapes that fit certain criteria.
    Diverse Learning Needs:
    Mathematics (2019)
    Grade(s): 8
    Accelerated
    All Resources: 0
    50. Apply the Pythagorean Theorem to determine unknown side lengths of right triangles, including real-world applications. [Grade 8, 28]
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Given real-world and mathematical problems in two and three dimensions, apply the Pythagorean Theorem in order to solve problems and justify solutions and solution paths for finding side lengths in right triangles within the problem contexts.
    Teacher Vocabulary:
    • Pythagorean Theorem
    • Right triangle
    • hypotenuse
    • leg
    Knowledge:
    Students know:
    • Pythagorean Theorem.
    • Appropriate labeling of a right triangle, (leg and hypotenuse).
    Skills:
    Students are able to:
    • Solve equations involving one variable and square root.
    • Represent real-world and mathematical contexts involving right triangles in a variety of formats (e.g., drawings, equations).
    • Justify solutions and solution paths using conceptual understandings and vocabulary related to the Pythagorean Theorem (e.g., right angle, hypotenuse).
    Understanding:
    Students understand that:
    • The properties of right triangles can be used to solve problems.
    Diverse Learning Needs: