Grades 9-12 Standards and Benchmarks Aligned with ITBS/ITED

Mathematics Curriculum Guide

Diocese of Sioux City

Grades 9-12

Standards and Benchmarks with Critical Objectives aligned with ITED

Standard 1: Understands and applies concepts of numbers and operations.

Benchmark 1: Describes the properties of numbers and number systems

Critical Objectives:

1.1.1.Understands properties of real numbers and its subsystems (ITED 9 ** 10 ** 11** 12**)

1.1.2.Knows the relationships among the subsystems of the real numbers system

1.1.3.Understands the type of equations that can and cannot be solved within each subsystem

1.1.4.Represents numbers within each subsystem (ITED 9 10 11 12)

1.1.5.Uses the appropriate number system representation for given problem situations (ITED 9 10 11 12)

1.1.6.Interprets numerical answers on a calculator or computer display

1.1.7.Explores new number systems, such as vectors and matrices

1.1.8.Knows what operation properties hold for operations with matrices and vectors

1.1.9.Determines what operation properties hold for matrix addition and multiplication

1.1.10.Calculates sums, differences, and products of matrices

1.1.11.Determines what operation properties hold for vector addition and subtraction

1.1.12.Calculates vector addition and multiplication

1.1.13.Uses the appropriate form of a rational number (fraction, decimal, percent) in computations (ITED 9*** 10 *** 11*** 12***)

1.1.14.Understands fractions, decimals, and percents can be expressed in various ways

1.1.15.Knows which rational number is most appropriate based on the context of the problem

1.1.16.Converts fractions, decimals, and percents to equivalent forms (ITED 9 10 11 12)

1.1.17.Identifies the appropriate rational number to use in a problem situation (ITED 9 10 11 12)

1.1.18.Uses the properties of roots, exponents in computations (ITED 9** 10** 11** 12**)

1.1.19.Knows squaring is the product of a whole number multiplied by itself

1.1.20.Knows squaring and square root are inverse operations

1.1.21.Knows an exponent tells how many times a base is used as a factor

1.1.22.Knows the rules for multiplying an dividing numbers with exponents

1.1.23.Knows the rules for raising a number with an exponent to a power

1.1.24.Finds the square root of any number

1.1.25.Writes exponents in expanded notation

1.1.26.Simplifies expressions containing exponents (ITED 9 10 11 12)

1.1.27.Writes expressions using scientific notation (ITED 9 10 11 12)

Benchmark 2: Understands the properties of operations

Critical Objectives:

1.2.1.Uses the properties of operations to simplify computations

1.2.2.Knows the inverse operations undo each other

1.2.3.Knows the identity properties of addition and multiplication

1.2.4.Knows the distributive, associative, and commutative properties of addition and multiplication

1.2.5.Knows squaring and square root are inverse operations

1.2.6.Simplifies equations using properties of operations

1.2.7.Explains the meaning of adding, subtracting, multiplying, and dividing integers

1.2.8.Uses inverse properties and relationships to solve problems

1.2.9.Finds the square root of any number

1.2.10.Uses the properties of operations to solve problems (ITED 9*** 10*** 11*** 12***)

1.2.11.Recognizes like terms

1.2.12.Understands rules for combining variables

1.2.13.Knows the order of operations

1.2.14.Knows the inverse relationship between properties of operations

1.2.15.Uses order of operation, including grouping symbols, to solve problems (ITED 9 10 11 12)

Benchmark 3: Computes fluently and makes reasonable estimates

Critical Objectives:

1.3.1.Uses a variety of operations on expressions containing complex numbers (ITED 9*** 10*** 11*** 12***)

1.3.2.Knows the algorithms of operations for complex numbers

1.3.3.Adds, subtracts, multiplies, divides, and simplifies expressions of complex numbers (ITED 9 10 11 12)

1.3.4.Uses estimation strategies for computing complex numbers (ITED 9*** 10*** 11*** 12***)

1.3.5.Knows when an estimate or an exact answer is more appropriate

1.3.6.Knows how to judge the reasonableness of an answer

1.3.7.Understands that rounding and estimation inherently add discrepancy

1.3.8.Understands appropriate places to round when estimating

1.3.9.Checks answers using estimation strategies (ITED 11 12)

1.3.10.Uses the appropriate places when estimating answers (ITED 9 10 11 12)

1.3.11.Makes reasonable estimates (ITED 9 10 11 12)

Standard 2: Understands and applies concepts of algebra and functions.

Benchmark 1:Represents patterns and relationships in a variety of ways

Critical Objectives:

2.1.1.Uses patterns and relationships to analyze mathematical situations (ITED 10 * 11* 12*)

2.1.2.Knows sequences and series can be used to model problems

2.1.3.Uses words, tables, and graphs and symbolic rules to describe patterns (ITED 10 11 12)

2.1.4.Describes the relationship of sequences symbolically

2.1.5.Generates formulas to describe a pattern

Benchmark 2: Describes the properties and characteristics of functions

Critical Objectives:

2.2.1.Describes functions and their properties using function notation (ITED 9* 10* 11* 12*)

2.2.2.Knows for every value in the domain of a function, there is one and only one corresponding value in the range

2.2.3.Understands the concept of a function as the correspondence between the elements of two sets

2.2.4.Understands the definition of a function: domain, range, function, non-function

2.2.5.Identifies the domain, range, and rule of a function

2.2.6.Describes functions and their properties using function notation

2.2.7.Recognizes the graphs of non-linear functions (ITED 10 11 12)

2.2.8.Describes from a graph the relationship between two variables (ITED 9 10 11 12)

2.2.9.Explores the properties of different classes of functions (ITED 9* 10** 11** 12**)

2.2.10.Knows the same function can be represented in a variety of forms

2.2.11.Knows the limitations of non-linear functions

2.2.12.Understands the concept of a class of functions

2.2.13.Identifies functions expressed numerically or algebraically as linear or non-linear (ITED 1011 12)

2.2.14.Recognizes properties of families of functions and how the values of parameters shape the graphs (ITED 9 10 11 12)

2.2.15.Compares properties of various functions to find common characteristics

Benchmark 3: Uses expressions and symbols to represent mathematical relationships

Critical Objectives:

2.3.1.Rewrites algebraic expression in equivalent forms (ITED 9*** 10*** 11** 12**)

2.3.2.Recognizes like terms

2.3.3.Knows rules to simplify expressions

2.3.4.Knows the order of operations

2.3.5.Understands the meaning of equivalent forms of expressions

2.3.6.Simplifies algebraic expressions by combining like terms and applying appropriate properties (ITED 9 10)

2.3.7.Uses dimensional analysis to organize conversions and computations (ITED 9 10 11 12)

2.3.8.Operates fluently on algebraic expressions by combining them and re-expressing them in alternate forms (ITED 9 10 11 12)

2.3.9.Manipulates polynomial expressions

2.3.10.Knows the various methods to factor polynomial expressions

2.3.11.Knows the properties of combining polynomials: addition, subtraction, multiplication, and division

2.3.12.Knows some polynomials cannot be solved precisely

2.3.13.Adds, subtracts, multiplies, and divides monomials and polynomials

2.3.14.Applies basic factoring techniques to second and simple third-degree polynomials

2.3.15.Simplifies fractions with polynomials in the numerator and denominator

2.3.16.Finds the number and type of roots of a polynomial equation

Benchmark 4: Uses models to represent mathematical situations

Critical Objectives:

2.4.1Uses a variety of methods to solve equations (ITED 9*** 10*** 11*** 12***)

2.4.2Recognizes like terms

2.4.3Understands rules for combining variables

2.4.4Knows the order of operations

2.4.5Knows the inverse relationship between addition and subtraction and multiplication and division

2.4.6Understands such operations as taking the opposite, finding the reciprocal, taking the square root

2.4.7Solves equations for a specified variable (ITED 9 10 11 12)

2.4.8Solves multi-step problems (ITED 9 10 11 12)

2.4.9Writes an equation for a specific problem (ITED 9 10 11 12)

2.4.10Uses a variety of methods to solve quadratic equations

2.4.11Knows the strengths and limitations of each method

2.4.12Understands the roots of a quadratic equation are given by the quadratic formula

2.4.13Knows the quadratic formula gives exact values

2.4.14Knows the discriminant can provide information about the roots of a quadratic equation

2.4.15Solves quadratic equations by graphing, factoring, completing the square, and the quadratic formula

2.4.16Analyzes problem situations to decide which method of solving quadratic equations is appropriate

2.4.17Uses a variety of methods to graph linear equations and inequalities (ITED 11* 12*)

2.4.18Knows how the slope and y-intercept of a graph are related to the equation

2.4.19Knows the algebraic significance of the parts of a graph

2.4.20Understands properties of graphs and relationships between a graph and its corresponding expression

2.4.21Knows the numerical value of the slope

2.4.22Knows the numerical value of y-intercept

2.4.23Graphs a linear equation and inequality

2.4.24Recognizes a linear equation and inequality from its graph (ITED 11 12)

2.4.25Finds the slope, x-intercept, and y-intercept of a line given its graph, its equation, or two points on the line (ITED 11 12)

2.4.26Writes equations in the General Line Equation format

2.4.27Identifies the dependent and independent variables

2.4.28Graphs a linear equation and computes the x- and y-intercepts

2.4.29Verifies a point lies on a line given an equation of the line

2.4.30Uses a variety of methods to solve systems of equations

2.4.31Knows the strengths and limitations of each method

2.4.32Knows whether a system of equations has one solution, no solutions, or infinitely many solutions

2.4.33Uses a graph to find the solution of a system of equations

2.4.34Uses substitution to solve a system of equations

2.4.35Uses elimination to solve a system of equations

Benchmark 5: Analyzes change in a variety of situations

Critical Objectives:

2.5.1Uses a variety of representations to analyze change (ITED 9* 10* 11** 12**)

2.5.2Understands change can be described mathematically

2.5.3Understands how rates of change can be described numerically and graphically

2.5.4Represents constant rates as the slope of a straight line graph (ITED 11 12)

2.5.5Interprets slope as the amount of one quantity (y) per unit amount of another (x) (ITED 11 12)

2.5.6Expresses rate of change graphically (ITED 11 12)

2.5.7Predicts rate of change from numerical and graphical data (ITED 9 10 11 12)

Standard 3: Understands and applies properties of geometry.

Benchmark 1: Uses properties of figures to verify relationships

Critical Objectives:

3.1.1.Describes figures by their characteristics

3.1.2.Knows the characteristics of figures

3.1.3.Describes geometric figures using their characteristics

3.1.4.Classifies figures by their discrete characteristics

3.1.5.Uses properties of lines to describe relationships between figures

3.1.6.Knows the properties of line segments

3.1.7.Knows the midpoint of a line segment divides the line into two equal parts

3.1.8.Knows how to find the length of line segments

3.1.9.Describes relationships of figures using line properties

3.1.10.Uses properties of angles to describe relationships between figures

3.1.11.Knows the properties of angles

3.1.12.Understands the relationship of angles when two parallel lines are cut by a transversal

3.1.13.Knows the relationship between the number of sides of a polygon and the measure of its interior angles

3.1.14.Describes relationships of figures using angle properties

3.1.15.Uses properties of circles to describe relationships between figures

3.1.16.Knows the properties of circles

3.1.17.Describes relationships of figures using properties of circles

3.1.18.Uses congruence and similarity to describe relationships among figures

3.1.19.Knows the properties of congruent and similar figures

3.1.20.Knows the difference between similarity and congruence

3.1.21.Knows the ratio for corresponding sides of similar triangles are the same

3.1.22.Recognizes figures as congruent or similar

3.1.23.Uses proportionality to show similarity

Benchmark 2: Uses coordinate geometry to verify relationships

Critical Objectives:

3.2.1.Uses the coordinate plane to show how geometric relationships correspond directly to algebraic concepts

3.2.2.Knows selecting the origins of a figure makes the calculations easier

3.2.3.Knows the formulas for distance, midpoint, and slope

3.2.4.Finds the distance and midpoints of segments in the coordinate plane

3.2.5.Compares the slopes and distances of line segments to verify a figure classification

3.2.6.Uses vertex-edge graphs to model and solve problems involving paths, networks, or relationships among a finite number of objects

Benchmark 3: Applies the concepts of transformation and symmetry

Critical Objectives:

3.3.1.Uses various representations to describe transformations

3.3.2.Knows that a transformation is a one-to-one correspondence

3.3.3.Knows the difference between the types of transformations

3.3.4.Knows how to describe position using matrices

3.3.5.Represents transformations using sketches, coordinates, vectors, function notation, and matrices

3.3.6.Identifies the types of transformations that make up a composition

3.3.7.Uses matrices to represent transformations

3.3.8.Describes figures in terms of their symmetry

3.3.9.Knows the difference among the types of symmetry

3.3.10.Identifies the type of symmetry in figures

Benchmark 4: Uses geometric reasoning to solve problems

Critical Objectives:

3.4.1.Uses proofs to build logical reasoning skills

3.4.2.Knows the difference between a definition, theorem, and a postulate

3.4.3.Knows the steps for completing a proof

3.4.4.Uses inductive and deductive reasoning to verify properties and relationships

3.4.5.Uses proofs to write convincing arguments

3.4.6.Establishes the validity of conjectures

3.4.7.Uses formulas for perimeter, area, and volume of various figures (ITED 9*** 10*** 11*** 12***)

3.4.8.Knows the formulas for perimeter, area, surface area, and volume

3.4.9.Knows how to solve a formula for any given variable

3.4.10.Computes the perimeter of various figures (ITED 9 10 11 12)

3.4.11.Computes the circumference of a circle (ITED 9 10 11 12)

3.4.12.Finds the length of an arc (ITED 11)

3.4.13.Computes the area of various figures (ITED9 10 11 12)

3.4.14.Computes the volume of various figures (ITED 9 10 11 12)

3.4.15.Uses properties of triangles to prove and verify relationships (ITED 9* 10*)

3.4.16.Knows the Pythagorean Theorem

3.4.17.Knows the properties of special triangles

3.4.18.Uses the Pythagorean Theorem and its converse to find segment lengths (ITED 9 10)

3.4.19.Uses the properties of special right triangles to solve problems

3.4.20.Uses trigonometric relationships to determine lengths and angle measurements

3.4.21.Knows the trigonometric ratios

3.4.22.Determines the appropriate trigonometric ratio for a specific problem

3.4.23.Uses trigonometric ratios to find lengths and angle measurements

Standard 4: Understands and applies concepts of measurement.

Benchmark 1: Understands measurable attributes and the process of measurement

Critical Objectives:

4.1.1.Uses indirect and derived measures to express measures of attributes (ITED 9* 10* 11* 12*)

4.1.2.Knows some attributes can only be measured indirectly

4.1.3.Knows some measures are derived measures

4.1.4.Knows some measurements of attributes are determined by a sequence of increasing approximations

4.1.5.Selects the appropriate indirect approach to solve a problem situation

4.1.6.Calculates attributes using indirect measures (ITED 9 10 11 12)

4.1.7.Uses rates as a measure of attributes

4.1.8.Uses increasing approximations to determine measures of attributes (ITED 9 10 11 12)

4.1.9.Applies the measurement process (ITED 9*)

4.1.10.Understands the process to be used when measuring attributes

4.1.11.Calculates error

4.1.12.Identifies the attribute to be measured

4.1.13.Chooses an appropriate unit (ITED 9)

4.1.14.Compares the unit with the object to be measured

Benchmark 2: Uses a variety of techniques for measurement

Critical Objectives:

4.2.1.Converts measurements between systems (ITED 9** 10** 11* 12**)

4.2.2.Knows units should be reported along with numerical values

4.2.3.Understands the meaning of equivalent forms of units

4.2.4.Knows units can be used to organize conversions

4.2.5.Uses dimensional analysis to organize conversions and computations (ITED 9 10 11 12)

4.2.6.Converts measurements from one system to another (ITED 1011 12)

4.2.7.Understands the importance of scale selection

4.2.8.Knows choices about scale and/or viewing windows help solve problems more effectively

4.2.9.Knows choices about scale and/or viewing windows can distort solutions

4.2.10.Uses the appropriate scale and/or viewing window to solve a problem

4.2.11.Uses formulas to find perimeter, area, and volume of various figures (ITED 9*** 10*** 11** 12**)

4.2.12.Knows the formulas for perimeter, circumference, area, surface area, and volume

4.2.13.Knows how to solve a formula for a given variable

4.2.14.Computes the perimeter of various figures (ITED 9 10 11 12)

4.2.15.Computes the circumference of a circle (ITED 11 12)

4.2.16.Computes the area of various figures (ITED 9 10 11 12)

4.2.17.Computes the volume of various figures (ITED 9 10 11 12)

4.2.18.Makes reasonable estimates about the accuracy of solutions (ITED 11* 12*)

4.2.19.Knows measurements of continuous quantities are approximations

4.2.20.Knows the degree of precision for a solution is dependent upon the accuracy of the measurement

4.2.21.Uses the appropriate degree of accuracy for reporting solutions (ITED 11 12)

4.2.22.Estimates answers to check accuracy of a solution

Standard 5: Understands and applies concepts of data analysis and probability.

Benchmark 1: Collects, organizes, and displays data to answer a question

Critical Objectives:

5.1.1.Gathers data to answer questions (ITED 9* 10*)

5.1.2.Knows different ways to collect a sample and the inherent bias in each

5.1.3.Knows how to formulate a questions that can be verified by data collection

5.1.4.Knows factors which add or detract from certainty of a data set

5.1.5.Formulates a question that can be answered by data collection

5.1.6.Collects data using a sampling technique

5.1.7.Compares data from two or more samples to determine how sampling can influence results

5.1.8.Uses knowledge of sampling, certainty, and data gathering to ascertain the certainty of the data (ITED 9 10)

5.1.9.Identifies possible bias in sampling

5.1.10.Represents data to convey results (ITED 9*** 10*** 11*** 12***)

5.1.11.Knows the basic types of charts, graphs, and tables

5.1.12.Knows different types of graphs convey different messages

5.1.13.Knows the criteria for selecting a graphical representation for a set of data

5.1.14.Knows types of errors in data representation that can lead to misinterpretation

5.1.15.Creates graphical representations appropriate for data type (ITED 9 10 11 12)

5.1.16.Identifies misleading representations of data

5.1.17.Extracts data from one graph and redisplays the data with a more appropriate representation (ITED 10 11 12)

5.1.18.Reads data from graphs (ITED 9 10 11 12)

5.1.19.Draws conclusions from data represented in a graph (ITED 9 10 11 12)

5.1.20.Makes predictions from data represented in a graph (ITED 9 10 11 12)

Benchmark 2: Uses statistical methods to describe data

Critical Objectives:

5.2.1.Describes data using central tendency and other statistical terms (ITED 11* 12*)

5.2.2.Knows when to remove outliers from a data set

5.2.3.Knows an outlier may represent an error in data collection or a significant variation in the data set

5.2.4.Knows the variability of a data set describes the "spread" of the numbers

5.2.5.Knows correlations can be causally related or incidental

5.2.6.Explains the mean's sensitivity to outliers

5.2.7.Determines the best measure of central tendency to describe a data set

5.2.8.Calculates and compares measures of central tendency in two data sets

5.2.9.Interprets representations of central tendency and variability (ITED 11 12)

5.2.10.Recognizes correlations as causal or incidental

5.2.11.Calculates correlations for a given data set

5.2.12.Describes factors that affect measures of central tendency (ITED 11* 12*)

5.2.13.Knows adding numbers to a data set can affect measures of central tendency

5.2.14.Knows an outlier is a data point that does not correspond with the trend in the data set

5.2.15.Knows outliers can affect measures of central tendency

5.2.16.Knows gaps are an area of the distribution where there are no data points

5.2.17.Knows a cluster is a group of localized points

5.2.18.Identifies situations in which the mean, mode, or median would contain the most relevant information

5.2.19.Calculates measures of central tendency with and without outliers (ITED 11 12)

5.2.20.Describes how the inclusion/exclusion of outliers affects measures of central tendency