instructional focus document algebra ii - wylie · functions will also be used to formulate and...

INSTRUCTIONAL FOCUS DOCUMENTAlgebra II

TITLE : Unit 02: Properties and Attributes of Functions SUGGESTED DURATION : 20 days

UNIT OVERVIEW

This unit bundles student expectations that address connections between functions and their characteristics, as well as connections between parent functions and their

transformations. The concept of the inverse of a function is also introduced and bundled within this unit.

Prior to this unit, in Algebra 1 Units 01, 04, 05, 09, 11, and 12, students experienced an in-depth study of linear and quadratic functions. Inverse (rational) and exponential

functions were also investigated in Algebra 1 Unit 1. In Geometry Units 05 and 07, linear and quadratic functions were applied in various pattern situations.

During this unit, students analyze functions by investigating their characteristics and representing them using various models. Students determine the domain (continuous or

discrete) and range as well as independent and dependent variables. Students identify functions as increasing or decreasing. Transformations are identified, described, and

applied to various parent functions. In addition, inverses of functions are explored by comparing and contrasting the function and its inverse using tables, graphs, and algebraic

representations.

After this unit, in Algebra 2 Units 03 – 10, the study of families of functions will be extended using data collection and analysis in problem situations. Each family of functionswill be studied in-depth by describing and applying the parent function, transformational effects of parameter changes, and characteristics of the function. Representative

functions will also be used to formulate and solve equations and inequalities in problem situations. Inverses will be addressed again when comparing and contrasting the

quadratic and square root functions as well as exponential and logarithmic functions.

Analysis of functions is foundational to high school and college mathematics. Focusing on the properties, attributes, and applications of functions using multiple

representations is emphasized in the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning C1; II. Algebraic Reasoning A1, C1, D1, D2; VI.

Statistical Reasoning B2; VII. Functions A1, B1, B2, C2; IX. Communication and Representation B1, B2, C1; and STAAR Readiness Standards: 2A.1A, 2A.1B, and 2A.4B.

Domain, range, inverses of functions, parent functions, and effects of parameter changes are addressed in STAAR Supporting Standards: 2A.4A, 2A.4C, 2A.7B, 2A.9A,

2A.9G, 2A.10A, and 2A.11B.

According to the National Council of Teachers of Mathematics (NCTM):

The Algebra Standard emphasizes relationships among quantities and the ways in which quantities change relative to one another. To think algebraically, one must be able to

understand patterns, relations, and functions; represent and analyze mathematical situations and structures using algebraic symbols; use mathematical models to represent

and understand quantitative relationships; and analyze change in various contexts… In high school, students create and use tables, symbols, graphs, and verbalrepresentations to generalize and analyze patterns, relations, and functions with increasing sophistication, and they convert flexibly among various representations. They

compare and contrast situations modeled by different types of functions, and they develop an understanding of classes of functions, both linear and nonlinear, and their

properties. (NCTM, 2002, p. 2-3)

In addition, Focus in High School Mathematics: Reasoning and Sense Making (2009) lists the key elements of reasoning and sense making with functions as being “usingmultiple representations of functions, modeling by using families of functions, and analyzing the effects of parameters” (NCTM, p. 41).

Last Updated 06/11/2014

Print Date 07/18/2014 Printed By Kerry Gain, EDUCATION SERVICE CENTERpage 1 of 25

National Council of Teachers of Mathematics. (2002). Navigating through algebra in grades 9 – 12. Reston, VA: National Council of Teachers of Mathematics, Inc.

National Council of Teachers of Mathematics. (2009). Focus in high school mathematics: Reasoning and sense making. Reston, VA: National Council of Teachers of

Mathematics, Inc.

Texas Education Agency. (2009). Texas college and career readiness standards. Austin, TX: Author.

PERFORMANCE ASSESSMENT(S)OVERARCHING CONCEPTS

UNIT CONCEPTSUNIT UNDERSTANDINGS

Algebra II Unit 02 PA 01

Analyze a function in a problem situation such as the

following:

The following data, comparing the mean height of

children in relation to their age, was collected in

Kalama, Egypt. Predict the mean height of a 21.5

month-old baby. Could you predict the height of a

21 year-old adult? Explain your reasoning.

Foundations for Functions –

Representations; Properties and

Attributes

Data relationships in problem situations can be interpreted through the

analysis of properties and attributes of functions and modeled using

multiple representations to make predictions and critical judgments in

terms of the problem situations.







Data from: United States Agency for International Development (USAID)

Create a graphic organizer that includes a table, graph,

verbal description, and algebraic generalization. Write an

analysis of the relationships between the

representations and a description of the attributes

(function/non-function, representative parent function,

domain/range, continuous/discrete,

increasing/decreasing, function notation, etc.). Use the

representations to make predictions and critical

judgments in the problem situation.

Standard(s): 2A.1A , 2A.1B ELPS.c.1C , ELPS.c.4J ,

ELPS.c.5B , ELPS.c.5G





about:/module/standards/0/107440/standard.ashx









Create a chart comparing effects of transformations. The

chart should include the name and equation of the

parent function, horizontal and vertical shifts, horizontal

and vertical stretch or compression, reflection, and a

composite graph of the parent function and translated

function. Formulate a sample graph of a quadratic

function that includes changes on all four parameters,

determine an algebraic generalization to represent the

quadratic function, and justify your reasoning.

Standard(s): 2A.4A , 2A.4B , 2A.7B , 2A.9A , 2A.10A ,

2A.11B ELPS.c.1C , ELPS.c.5B , ELPS.c.5F

Algebra and Geometry – Parent

Functions

Quadratic and Square Root Functions

– Effects of Parameter Changes

Rational Functions – Effects of

Parameter Changes

Exponential and Logarithmic Functions

– Effects of Parameter Changes

Changes in the parameters of parent functions have specific effects on

the equation of the function and the graph in the coordinate plane.


Compare and contrast a function and its inverse given

various representations such as the following:

graph

Algebra and Geometry – Inverses of

Functions

Quadratic and Square Root Functions

– Inverses of Functions

The graph of a function and its inverse reflect across the y = x line,

and the elements of the ordered pairs are reversed; the x and y

variables are interchanged in the algebraic generalizations.
















table

equation

Create a graphic organizer that includes a table of

values, a graph, and an algebraic generalization for both

the function and its inverse. Write a summary comparing

the domain and range, tables, graphs, and algebraic

generalizations of the function and its inverse.

Standard(s): 2A.4C , 2A.9G ELPS.c.1C , ELPS.c.5B ,

ELPS.c.5G










MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

MISCONCEPTIONS:

Some students may think that the domain and range of a function and the domain and range of the problem situation are always equivalent.

Some students may think that the inverse of a function means to take the opposite sign or reciprocal of the function instead of switching the independent and dependent

variables.

UNDERDEVELOPED CONCEPTS:

Some students may not have an understanding of the distinctions between discrete and continuous domains or how to represent discrete domains graphically and

symbolically.

UNIT VOCABULARY

Functions – relations in which each element of the input (domain) is paired with exactly one element of the output (range)

Parent functions – set of basic functions from which related functions are derived by transformations

Relation – a set of ordered pairs (pairing of a set of inputs (domain) with a set of outputs (range)

Transformations – general term for ways to manipulate the shape or position of a graph; types of transformation include translation (shifts), reflection, and dilation

(compression and stretch); specific parameter changes on parent functions cause certain types of transformations on the graph of the function.





SYSTEM RESOURCES OTHER RESOURCES

Algebra II Transition Alignment Guide (TAG) Tool Algebra 2 End of Course Success: Vocabulary Reporting Category 1 – On the Move,On Your Own: Putting It All Together, Keep It Moving 1, Keep It Moving 2

MTR Grades 9 – 12: Tab 4, Algebra II: Understanding

Functions http://www.txar.org/professional_dev/archived/training/materials/MTR/9-

12/pdfs/tab4.pdf

TEA STAAR Mathematics

Resources: http://www.tea.state.tx.us/student.assessment/staar/math/

TEA STAAR Released Test

Questions: http://www.tea.state.tx.us/student.assessment/staar/testquestions/

TEXTEAMS: Part I: Algebra II/Precalculus Institute: I – Foundations of Functions; 1.Foundations, 1.1 Transformations, Act. 1 (Explore Transformations), Act. 2 (Domains),

Act. 3 (Transformations with Technology), Act. 4 (Summarize Your Findings); 2. Applying

Transformations, 2.1 Applications, Act. 1 (Good Viewing Windows), Act. 2 (Friendly

Viewing Windows)

TEKS UNIT LEVEL SPECIFICITY

Legend:

Bold black text in italics: Knowledge and

Skills Statement (TEKS)

Bold black text: Student Expectation (TEKS)

Bolded red text in italics: Student

Expectation identified by TEA as a

Readiness Standard for STAAR

Bolded green text in italics: Student

Legend:

Blue text: Supporting Information / Clarifications from TCMPC (Specificity)

Blue text in italics: Unit-specific clarification

Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards

(TxCCRS)

TEKS#

SE#





about:/module/content/search/item/676360/viewdetail.ashx

http://www.txar.org/professional_dev/archived/training/materials/MTR/9-12/pdfs/tab4.pdf

http://www.tea.state.tx.us/student.assessment/staar/math/

http://www.tea.state.tx.us/student.assessment/staar/testquestions/

TEKS UNIT LEVEL SPECIFICITYTEKS#

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Expectation identified by TEA as a

Supporting Standard for STAAR

Strike-through: Indicates portions of the Student

Expectation that are not included in this unit but

are taught in previous or future unit(s)

2A.1 Foundations for functions. The student uses

properties and attributes of functions and applies

functions to problem situations. The student is

expected to:

2A.1A Identify the mathematical domains and ranges of

functions and determine reasonable domain and

range values for continuous and discrete

situations.

Readiness Standard

Use, Apply, Identify, Determine

REASONABLE MATHEMATICAL DOMAINS AND RANGES

Including, but not limited to:

Comparison of discrete and continuous domain

Representations of domain and range with and without technology

Tables

Graphs

Verbal descriptions

Algebraic generalizations (including equation and function notation)

Connections among the representations

Notation of domain and range including set builder notation

Lists of domain and range in sets (including f(x) for y)

Verbal

Ex: Domain is all real numbers; domain is all real numbers greater than five.





about:/module/standards/Tools/Browse?StandardId=107439



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Ex: Range is all real numbers less than zero; range is all real numbers.

Symbolic

Ex: domain (e.g., x | x > 0; x | -3 < x < 4; x | x R; Ø)

Ex: range (e.g., y | y ≤ 0; y | -7 < y ≤ 4; f(x) | -5 < f(x) ≤ 2; y | y R; Ø)

Determination of scales for graphs and windows on graphing calculators using domain and

range

Contextual domain and range of real-world problem situation

Specific range values at given domain values from various representations (e.g., tables, graphs,

or algebraic generalizations)

Specific domain values at given range values from various representations (e.g., tables, graphs,

or algebraic generalizations)

Domain and range of the problem situation versus the domain and range of the mathematical

function

Note:

In Algebra 1, students determine domain and range using various representations of functions.

TxCCRS Note:

VII. Functions B1 – Understand and analyze features of a function.

2A.1B Collect and organize data, make and interpret

scatterplots, fit the graph of a function to the data,

interpret the results, and proceed to model,

predict, and make decisions and critical

judgments.

Readiness Standard

Collect, Organize

DATA


Data sets and real-world problem situations

Data collection and analysis with and without the use of technology (e.g., CBR/CBL, graphing







SE#

calculators, and computers)

Ex: measurement activities, experiments, simulations, Internet, etc.

Representations of data with and without technology

Tables

Graphs including scatterplots

Verbal descriptions


Use, Make, Interpret, Fit, Model, Apply

SCATTERPLOTS AND REPRESENTATIVE FUNCTIONS FOR DATA



General trends in data





Tables

Graphs, including scatterplots

Verbal descriptions


Parent functions as representations of scatterplots of data with and without technology

Linear

Quadratic

Square root

Rational

Exponential






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Logarithmic

Absolute value

Linear or nonlinear scatterplots using graphing technology

Model, Predict, Make

DECISIONS AND CRITICAL JUDGMENTS IN PROBLEM SITUATIONS



General trends in the data with and without technology





Tables

Graphs including scatterplots

Verbal descriptions


Parent functions as representations of scatterplots of data with and without technology

Linear

Quadratic

Square root

Rational

Exponential

Logarithmic

Absolute value

Linear or nonlinear scatterplots using graphing technology

Predictions and critical judgments in terms of the scatterplots and representative functions






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Note:

In Algebra 1, students collect and analyze data using various representations, which are then

used to make predictions and critical judgments.

TxCCRS Note:

II. Algebraic Reasoning D1 – Interpret multiple representations of equations and relationships.II. Algebraic Reasoning D2 – Translate among multiple representations of equations and relationships.III. Geometric Reasoning C2 – Make connections between geometry, statistics, and probability.VI. Statistical Reasoning A1 – Plan a study.VI. Statistical Reasoning B1 – Determine types of data.VI. Statistical Reasoning B2 – Select and apply appropriate visual representations of data.VI. Statistical Reasoning B4 – Describe patterns and departure from patterns in a set of data.VI. Statistical Reasoning C1 – Make predictions and draw inferences using summary statistics.VI. Statistical Reasoning C2 – Analyze data sets using graphs and summary statistics.VI. Statistical Reasoning C3 – Analyze relationships between paired data using spreadsheets,graphing calculators, or statistical software.

VII. Functions A2 – Recognize and distinguish between different types of functions.VII. Functions B2 – Algebraically construct and analyze new functions.VII. Functions C1 – Apply known function models.VII. Functions C2 – Develop a function to model a situation.VIII. Problem Solving and Reasoning B2 – Use various types of reasoning.IX. Communication and Representation B1 – Model and interpret mathematical ideas and conceptsusing multiple representations.

IX. Communication and Representation B2 – Summarize and interpret mathematical informationprovided orally, visually, or in written form within the given context.

IX. Communication and Representation C1– Communicate mathematical ideas, reasoning, and theirimplications using symbols, diagrams, graphs, and words.

X. Connections A1 – Connect and use multiple strands of mathematics in situations and problems.







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2A.4 Algebra and geometry. The student connects

algebraic and geometric representations of

functions. The student is expected to:

2A.4A Identify and sketch graphs of parent functions,

including linear (f (x) = x), quadratic (f (x) = x2),

exponential (f (x) = ax), and logarithmic (f (x) =

logax) functions, absolute value of x (f (x) = |x|),

square root of x (f (x) = √x), and reciprocal of x (f(x) = 1/x).

Supporting Standard

Connect, Identify, Sketch

PARENT FUNCTIONS AND THEIR GRAPHS


General forms of parent functions (including equation and function notation)

Linear: f(x) = x

Quadratic: f(x) = x²

Exponential: f(x) = ax

Logarithmic: f(x) = logax

Absolute value: f(x) = |x|

Square root: f(x) =√x

Rational (reciprocal of x): f(x) =

Representations of parent functions with and without technology

Tables

Graphs

Verbal descriptions


Connections between representations

Characteristics of parent functions with and without technology

Ex: domain and range, rate of change, increasing/decreasing, vertex points, end points,

intercepts, symmetry, asymptotes, maximum/minimum, end behavior, etc.

Connections between transformed functions to the respective parent function with and without








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technology

Similarities and differences in the parent functions

Identification of relations as functions (TxCCRS)

Note:

In Algebra 1, students identified the characteristics of the linear and quadratic parent functions.

TxCCRS Note:

III. Geometric Reasoning C1 – Make connections between geometry and algebra.VII. Functions A1 – Recognize whether a relation is a function.VII. Functions A2 – Recognize and distinguish between different types of functions.VII. Functions B1 – Understand and analyze features of a function.X. Connections A1 – Connect and use multiple strands of mathematics in situations and problems.

2A.4B Extend parent functions with parameters such as

a in f (x) = a/x and describe the effects of the

parameter changes on the graph of parent

functions.

Readiness Standard

Extend, Describe

PARAMETER CHANGES ON PARENT FUNCTIONS


General forms of parent functions (including equation and function notation)

Linear: f(x) = x

Quadratic: f(x) = x²

Exponential: f(x) = ax

Logarithmic: f(x) = logax

Absolute value: f(x) = |x|

Square root: f(x) =√x

Rational (reciprocal of x): f(x) =







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Representations of parent functions with and without technology

Graphs

Verbal descriptions


Changes in parameters a, h, and k on the graph of the parent function:

Effects of changes in a on the graph of the parent function with and without technology

a ≠ 0|a| > 1, the graph stretches vertically

0 < |a| < 1, the graph compresses vertically

Opposite of a reflects vertically over the horizontal axis (x-axis)

Effects of changes in h on the graph of the parent function with and without technology

h = 0, no horizontal shift

Horizontal shift left or right

Effects of changes in k on the graph of the parent function with and without technology

k = 0, no vertical shift

Vertical shift up or down

Determination of parameter changes given a graphical or algebraic representation

Determination of a graphical representation given the algebraic representation or parameter

changes

Determination of an algebraic representation given the graphical representation or parameter

changes

Connections between the critical attributes of transformed functions and their respective parent

functions

Descriptions of the effects on the domain and range by the parameter changes

Effects of parameter changes in mathematical situations

Effects of parameter changes in problem situations






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Note:

In Algebra 1, students extended parameters on vertical shifts, stretches, and compressions for

linear and quadratic functions.

TxCCRS Note:

III. Geometric Reasoning B1 – Identify and apply transformations to figures.III. Geometric Reasoning C1 – Make connections between geometry and algebra.VII. Functions B1 – Understand and analyze features of a function.VII. Functions B2 – Algebraically construct and analyze new functions.VII. Functions C1 – Apply known function models.X. Connections A1 – Connect and use multiple strands of mathematics in situations and problems.

2A.4C Describe and analyze the relationship between a

function and its inverse.

Supporting Standard

Describe, Analyze

RELATIONSHIPS BETWEEN FUNCTIONS AND THEIR INVERSES


Comparison of independent and dependent variables of a function and its inverse

Comparison of domain and range of a function and its inverse

Symmetry about y = x

Identification of functions and non-functions

Comparison of functions and their inverses using multiple representations with and without

technology

Tables

Graphs

Verbal descriptions


Inverse relationships







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Quadratic and square root functions

Exponential and logarithmic functions

TxCCRS Note:

III. Geometric Reasoning C1 – Make connections between geometry and algebra.VII. Functions A1 – Recognize whether a relation is a function.VII. Functions B1 – Understand and analyze features of a function.X. Connections A1 – Connect and use multiple strands of mathematics in situations and problems.

2A.7 Quadratic and square root functions. The student

interprets and describes the effects of changes in

the parameters of quadratic functions in applied

and mathematical situations. The student is

expected to:

2A.7B Use the parent function to investigate, describe,

and predict the effects of changes in a, h, and k

on the graphs of y = a (x - h)2 + k form of a

function in applied and purely mathematical

situations.

Supporting Standard

Interpret, Use, Investigate, Describe, Predict

PARAMETER CHANGES ON QUADRATIC FUNCTIONS IN APPLIED AND MATHEMATICAL

SITUATIONS


Comparison of quadratic parent function y = x2 to transformations reflected in y = a(x - h)2 + k

with and without technology

Representations of parent function and related function with and without technology

Tables

Graphs

Verbal descriptions


Changes in parameters a, h, and kon the graph of the parent function, y = a(x - h)2 + k








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changes


changes


functions

Descriptions of the effects on the domain and range by the parameter changes



TxCCRS Note:

III. Geometric Reasoning B1 – Identify and apply transformations to figures.VII. Functions B1 – Understand and analyze features of a function.

2A.9 Quadratic and square root functions. The student

formulates equations and inequalities based on

square root functions, uses a variety of methods to







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solve them, and analyzes the solutions in terms of

the situation. The student is expected to:

2A.9A Use the parent function to investigate, describe,

and predict the effects of parameter changes on

the graphs of square root functions and describe

limitations on the domains and ranges.

Supporting Standard

Use, Investigate, Describe, Predict

GRAPHS AND EFFECTS OF PARAMETER CHANGES ON SQUARE ROOT FUNCTIONS


Comparison of square root parent function y=√x to transformations reflected in y=a√(xh) + k

with and without technology


Tables

Graphs

Verbal descriptions


Changes in parameters a, h, and k on the graph of the parent function, y=a√(xh)+ k


















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changes


changes


functions

Descriptions of the effects on the domain and range and limitations of domain by the parameter

changes



TxCCRS Note:

III. Geometric Reasoning B1 – Identify and apply transformations to figures.VII. Functions B1 – Understand and analyze features of a function.VII. Functions B2 – Algebraically construct and analyze new functions.

2A.9G Connect inverses of square root functions with

quadratic functions.

Supporting Standard

Connect, Analyze

SQUARE ROOTS AS INVERSE FUNCTIONS OF QUADRATICS


Inverse relationships

Quadratic functions

Square root functions

Characteristics of inverse relationships with and without technology

Domain and range switched in table

Reflection in the y = x line on the graph

Algebraic representations (switching x and y and solving for y)

Comparison of functionality of the original function and its inverse (horizontal and vertical line







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tests) with and without technology

TxCCRS Note:

VII. Functions A1 – Recognize whether a relation is a function.VII. Functions B1 – Understand and analyze features of a function.

2A.10 Rational functions. The student formulates

equations and inequalities based on rational

functions, uses a variety of methods to solve them,

and analyzes the solutions in terms of the situation.

The student is expected to:

2A.10A Use quotients of polynomials to describe the

graphs of rational functions, predict the effects of

parameter changes, describe limitations on the

domains and ranges, and examine asymptotic

behavior.

Supporting Standard

Use, Examine, Describe, Analyze, Predict

EFFECTS OF PARAMETER CHANGES ON RATIONAL FUNCTIONS


Comparison of rational parent function y = to transformations reflected in


Tables

Graphs

Verbal descriptions


Changes in parameters a, h, and k on the graph of the parent function


a ≠ 0








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|a| > 1, the graph stretches vertically











changes


changes


functions

Descriptions of the effects on the domain and range and limitations of domain by the parameter

changes



TxCCRS Note:

III. Geometric Reasoning B1 – Identify and apply transformations to figures.VII. Functions B1 – Understand and analyze features of a function.VII. Functions B2 – Algebraically construct and analyze new functions.

2A.11 Exponential and logarithmic functions. The student

formulates equations and inequalities based on

exponential and logarithmic functions, uses a

variety of methods to solve them, and analyzes the







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solutions in terms of the situation. The student is

expected to:

2A.11B Use the parent functions to investigate, describe,

and predict the effects of parameter changes on

the graphs of exponential and logarithmic

functions, describe limitations on the domains

and ranges, and examine asymptotic behavior.

Supporting Standard

Use, Investigate, Describe, Predict, Examine, Analyze

GRAPHS AND EFFECTS OF PARAMETER CHANGES ON EXPONENTIAL AND LOGARITHMIC

FUNCTIONS


Comparison of exponential parent function f(x)=bx to transformations reflected in f(x)=a*bx-h +k

Comparison of logarithmic parent function f(x)=logbx to transformations reflected in f(x)= alogb

(x-h) +k


Tables

Graphs

Verbal descriptions


Changes in parameters a, h, and k on the graphs of the parent function, f(x)=a*bx-h +k

Changes in parameters a, h, and kon the graphs of the parent function, f(x)= alogb (x-h) +k















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changes


changes


functions

Descriptions of the effects on the domain and range and limitations of domain (logarithms) by

the parameter changes



Note:

In the exponential and logarithmic parent functions the a, used as the exponent and log base,

was changed to b, so it would not be confused with the a parameter used in transformations of

parent functions.

TxCCRS Note:

III. Geometric Reasoning B1 – Identify and apply transformations to figures.VII. Functions B1 – Understand and analyze features of a function.VII. Functions C1 – Apply known function models.





The English Language Proficiency Standards (ELPS), as required by 19 Texas Administrative Code, Chapter 74, Subchapter A, §74.4, outline English languageproficiency level descriptors and student expectations for English language learners (ELLs). School districts are required to implement ELPS as an integral part of

each subject in the required curriculum.

School districts shall provide instruction in the knowledge and skills of the foundation and enrichment curriculum in a manner that is linguistically accommodated

commensurate with the student’s levels of English language proficiency to ensure that the student learns the knowledge and skills in the required curriculum.

School districts shall provide content-based instruction including the cross-curricular second language acquisition essential knowledge and skills in subsection (c) of the

ELPS in a manner that is linguistically accommodated to help the student acquire English language proficiency.

http://ritter.tea.state.tx.us/rules/tac/chapter074/ch074a.html#74.4

Choose appropriate ELPS to support instruction.

ELPS# SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS.






http://ritter.tea.state.tx.us/rules/tac/chapter074/ch074a.html#74.4

instructional focus document algebra ii - wylie · functions will also be used to formulate and...

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