INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 03: Linear Functions, Equations, and Inequalities
SUGGESTED DURATION : 13 days
UNIT OVERVIEW
This unit bundles student expectations that investigate and explore the characteristics and applications of linear functions, equations, and inequalities. In addition, the linear
parent function is introduced and effects of parameter changes on the graph of the parent function are explored.
Prior to this unit, in Algebra 1 Units 02 – 04, students completed an in­depth study of the characteristics and properties of linear functions including transformation of the linear
parent function. Students also explored multiple representations of linear functions, equations, and inequalities and solved problem situations modeled by linear functions.
During this unit, students analyze characteristics and applications of linear functions in real-life situations. Students determine the strength of a linear model for a set of data
using statistical analysis, including the linear correlation coefficient. Linear functions are explored and modeled using various representations, and characteristics such as
slope and intercepts, are determined and analyzed. Equations and inequalities are formulated, modeled, and solved by various methods to make predictions in terms of
problem situations.
After this unit, in Algebra 2 Units 04 – 10 and 12, and in subsequent mathematics courses, students will apply similar concepts to investigate additional types of functions and
equations.
Analysis of linear 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): II. Algebraic Reasoning A1, B1, C1, D1, D2; Geometric Reasoning B1, C1,
C2; Measurement Reasoning D1; VI. Statistical Reasoning A1, B1, B2, B4, C1, C2, C3; VII. Functions A1, A2, B1, B2, C1, C2; VII. Problem Solving and Reasoning B2; IX.
Communication and Representation B1, B2, C1; Connections A1; and STAAR Readiness Standards: 2A.1A, 2A.1B, and 2A.4B. Simplifying expressions and solving equations
is addressed in STAAR Supporting Standard: 2A.2A; and the linear parent function is addressed in STAAR Supporting Standard: 2A.4A.
According to the National Council of Teachers of Mathematics (2000), students should have an opportunity to build on these earlier experiences, both deepening their
understanding of relations and functions and expanding their repertoire of familiar functions. High school algebra should provide students with insights into mathematical
abstraction and structure. High school students’ algebra experience should enable them to create and use tabular, symbolic, graphical, and verbal representations and to
analyze and understand patterns, relations, and functions with more sophistication than in the middle grades. Students should develop an understanding of the algebraic
properties that govern manipulation of symbols in expressions, equations, and inequalities.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency. (2009). Texas college and career readiness standards. Austin, TX: Author.
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page 1 of 17 INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 03: Linear Functions, Equations, and Inequalities
PERFORMANCE ASSESSMENT(S)
Algebra II Unit 03 PA 01
Analyze a problem situation that can be represented by
a linear function such as the following:
The number of dollars per month it costs you to
own a car is a function of the number of
kilometers a month you drive it. According to
Time magazine, the cost varies linearly with the
distance. The article states that driving 300 km
costs $366 per month, but going 1,500 km in a
month would cost $510. Predict the monthly cost
of owning a car for drivers who go 500 km/month,
1,000 km/month, and 2,000 km/month. About
how far could you drive in a month without
exceeding a monthly cost of $600?
SUGGESTED DURATION : 13 days
OVERARCHING CONCEPTS
UNIT CONCEPTS
UNIT UNDERSTANDINGS
Foundations for Functions –
Representations; Properties and
Attributes
Slope, intercepts, and the equation of the line can be determined from
various representations of data.
Foundations for Functions –
Representations; Properties and
Attributes
Changes in slope and the y-intercept have specific effects on the
graph of the linear parent function and in the problem situation.
Algebra and Geometry – Parent
Functions; Effects of Parameter
Changes
Problem situations or samples of collected data involving linear
relationships can be represented by various methods. These
representations can be used to predict values over an appropriate
domain and range.
Create a graphic organizer that includes a representative
function, table, graph, and specific characteristics
(independent/dependent, domain/range, slope, yintercept, x-intercept, and their meaning in the problem
situation). Write a summary of the effects of changing
the slope and y-intercept on the graph of the linear
parent function and in the problem situation. Use the
representations to make predictions and critical
judgments in the problem situation.
Standard(s): 2A.1A , 2A.1B , 2A.4A , 2A.4B ELPS.c.1C
Last Updated 07/15/2014
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page 2 of 17 INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 03: Linear Functions, Equations, and Inequalities
PERFORMANCE ASSESSMENT(S)
SUGGESTED DURATION : 13 days
OVERARCHING CONCEPTS
UNIT CONCEPTS
UNIT UNDERSTANDINGS
, ELPS.c.5B , ELPS.c.5G
Algebra II Unit 03 PA 02
Write a two variable inequality to describe a problem
situation such as the following:
Foundations for Functions –
Representations; Properties and
Attributes; Equations; Inequalities
Problem situations can be modeled using two-variable inequalities,
and the solution can be represented graphically on a coordinate plane.
In a chemistry experiment, students kept track of
the temperature of a solution as it heated over a
period of time. That data is displayed in the table
below. About how long would it take to heat the
solution to a temperature of 257°F? Later in the
semester, a second chemistry experiment will be
conducted where the solution attains lower
temperatures when heated over the same period
of time.
Represent the solution graphically on a coordinate
plane. Use the graphical representation to write a
summary of the description of the solution in terms of
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page 3 of 17 INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 03: Linear Functions, Equations, and Inequalities
PERFORMANCE ASSESSMENT(S)
SUGGESTED DURATION : 13 days
OVERARCHING CONCEPTS
UNIT CONCEPTS
UNIT UNDERSTANDINGS
the problem situation. Display all results using a graphic
organizer.
Standard(s): 2A.1A , 2A.1B , 2A.4A , 2A.4B ELPS.c.1C
, ELPS.c.5B , ELPS.c.5G
Algebra II Unit 03 PA 03
Analyze problem situations that can be represented
algebraically by equations or inequalities such as the
following:
Foundations for Functions –
Representations; Properties and
Attributes; Equations; Inequalities
Equations and inequalities can be written to represent problem
situations and solved over an appropriate domain. The reasonableness
of the solution can be justified in terms of the problem situation.
The length of a rectangular garden is seven feet
longer than its width. The perimeter of the garden
is 38 feet. Find the length and width of the garden
to the nearest foot.
Susie’s math quiz scores are 73, 75, 89, and 91.
What is the lowest score she can get on the last
quiz and still have an average of at least 85?
Create a graphic organizer for each problem situation
that includes formulating a representative equation or
inequality, solving the equation or inequality over an
appropriate domain, and interpreting the reasonableness
of the solution in terms of the problem situation.
Standard(s): 2A.1A , 2A.2A ELPS.c.1C , ELPS.c.4J ,
ELPS.c.5B
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page 4 of 17 INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 03: Linear Functions, Equations, and Inequalities
SUGGESTED DURATION : 13 days
MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS
None Identified
UNIT VOCABULARY
Correlation coefficient – measure of the strength and direction of the linear relationship between two variables in a set of data, also known as the r-value
Line of regression – line of best fit to the data points on a scatterplot mathematically calculated using the least­squares method (minimizes the sum of the squares of
the distances between the line and the data points)
Trend line – the line that best fits the data points on a scatterplot when adjusting the parameters on the parent function
SYSTEM RESOURCES
Algebra II Transition Alignment Guide (TAG) Tool
OTHER RESOURCES
Fostering Algebraic Thinking: Introduction and Analyzing Written Student Work,
Session 2 – Sum of Consecutive Numbers; Asking Questions of Students, Session 1 –
The Candles Problem, Session 2 – Towering Numbers, Session 3 – Toothpick Squares;
Documenting Patterns of Student Thinking, Session 2 – Graph Stories
Maximizing Algebra 2 Performance: Explore/Explain 3: Absolute Value Functions –
Part 4 (Solving Absolute Value Equations and
Inequalities)http://ritter.tea.state.tx.us/math/training/materials/Algebra_II/index.htm
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: I – Foundations of Functions; 2. Applying
Transformations, 2.2 Airlines, Act. 1 (Trend Lines), Act. 2 (Another Trend Line); 3. Making
Connections, 3.2 Stays the Same, Act. (Solving Equations Step by Step Graphically)
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Algebra II
TITLE : Unit 03: Linear Functions, Equations, and Inequalities
SUGGESTED DURATION : 13 days
The phase 2 College Readiness English Language Arts and Reading vertical alignment team found that the College Readiness Standards in English
Language Arts and Reading are well aligned with the Texas Essential Knowledge and Skills.
TEKS#
TEKS
SE#
Legend:
Bold black text in italics: Knowledge and
Skills Statement (TEKS)
UNIT LEVEL SPECIFICITY
Legend:
Blue text: Supporting Information / Clarifications from TCMPC (Specificity)
Blue text in italics: Unit-specific clarification
Bold black text: Student Expectation (TEKS)
Bolded red text in italics: Student
Expectation identified by TEA as a
Readiness Standard for STAAR
Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards
(TxCCRS)
Bolded green text in italics: Student
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
Use, Apply, Identify, Determine
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Algebra II
TITLE : Unit 03: Linear Functions, Equations, and Inequalities
SUGGESTED DURATION : 13 days
TEKS#
SE#
TEKS
range values for continuous and discrete
situations.
Readiness Standard
UNIT LEVEL SPECIFICITY
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.
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 Î R}; Ø)
Ex: range (e.g., {y | y ≤ 0}; {y | -7 < y ≤ 4}; { f(x) | -5 < f(x) ≤ 2}; {y | y Î 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
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page 7 of 17 INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 03: Linear Functions, Equations, and Inequalities
SUGGESTED DURATION : 13 days
TEKS#
SE#
TEKS
UNIT LEVEL SPECIFICITY
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
Including, but not limited to:
Data sets and real-world problem situations
Data collection and analysis with and without the use of technology (e.g., CBR/CBL, graphing
calculators, and computers)
Ex: measurement activities, experiments, simulations, Internet, etc.
Representations of data with and without technology
Tables
Graphs including scatterplots
Verbal descriptions
Algebraic generalizations (including equation and function notation)
Use, Make, Interpret, Fit, Model, Apply
SCATTERPLOTS AND REPRESENTATIVE FUNCTIONS FOR DATA
Including, but not limited to:
Data sets and real-world problem situations
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Algebra II
TITLE : Unit 03: Linear Functions, Equations, and Inequalities
SUGGESTED DURATION : 13 days
TEKS#
SE#
TEKS
UNIT LEVEL SPECIFICITY
General trends in data
Data collection and analysis with and without the use of technology (e.g., CBR/CBL, graphing
calculators, and computers)
Ex: measurement activities, experiments, simulations, Internet, etc.
Representations of data with and without technology
Tables
Graphs, including scatterplots
Verbal descriptions
Algebraic generalizations (including equation and function notation)
Parent functions as representations of scatterplots of data with and without technology
Linear
Linear scatterplots using graphing technology
Positive linear correlation
Negative linear correlation
No linear correlation
Model, Predict, Make
DECISIONS AND CRITICAL JUDGEMENTS IN PROBLEM SITUATIONS
Including, but not limited to:
Data sets and real-world problem situations
General trends in the data with and without technology
Data collection and analysis with and without the use of technology (e.g., CBR/CBL, graphing
calculators, and computers)
Ex: measurement activities, experiments, simulations, Internet, etc.
Representations of data with and without technology
Tables
Graphs including scatterplots
Last Updated 07/15/2014
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Algebra II
TITLE : Unit 03: Linear Functions, Equations, and Inequalities
SUGGESTED DURATION : 13 days
TEKS#
SE#
TEKS
UNIT LEVEL SPECIFICITY
Verbal descriptions
Algebraic generalizations (including equation and function notation)
Parent functions as representations of scatterplots of data with and without technology
Linear
Linear scatterplots using graphing technology
Positive linear correlation
Negative linear correlation
No linear correlation
Predictions and critical judgments in terms of the scatterplots and representative functions
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.
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Algebra II
TITLE : Unit 03: Linear Functions, Equations, and Inequalities
SUGGESTED DURATION : 13 days
TEKS#
TEKS
SE#
UNIT LEVEL SPECIFICITY
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 concepts
using multiple representations.
IX. Communication and Representation B2 – Summarize and interpret mathematical information
provided orally, visually, or in written form within the given context.
IX. Communication and Representation C1– Communicate mathematical ideas, reasoning, and their
implications using symbols, diagrams, graphs, and words.
X. Connections A1 – Connect and use multiple strands of mathematics in situations and problems.
2A.2
Foundations for functions. The student understands
the importance of the skills required to manipulate
symbols in order to solve problems and uses the
necessary algebraic skills required to simplify
algebraic expressions and solve equations and
inequalities in problem situations. The student is
expected to:
2A.2A
Use tools including factoring and properties of
exponents to simplify expressions and to
transform and solve equations.
Supporting Standard
Use
MATHEMATICAL TOOLS
Including, but not limited to:
Algebraic properties of equations and inequalities
Ex: commutative, associative, distributive, identities, inverses, etc.
Manipulate, Simplify
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Algebra II
TITLE : Unit 03: Linear Functions, Equations, and Inequalities
SUGGESTED DURATION : 13 days
TEKS#
SE#
TEKS
UNIT LEVEL SPECIFICITY
EXPRESSIONS
Including, but not limited to:
Mathematical expressions
Evaluation of expressions when given values for variables
Simplification of expressions using properties of algebra
Algebraic generalization of a data set or problem situation
Functional expressions
Equation notation
Ex: If (2, y) is a solution to the equation, find y if 2x – 3y = 7.
Ex: Given the equation y = 2x + 3, where y is dependent on x, what are the
dependent values if x Î {­4, 0, 2, 12}?
Function notation
Ex: If f(x) = 2x – 1, find f(2).
Ex: If f(x) = 2x – 1 and f(x) = 5, find x.
Function values using technology (e.g., the table or trace features on the graphing
calculator)
Tools (e.g., algebraic properties)
Manipulate, Transform, Solve
EQUATIONS AND INEQUALITIES
Including, but not limited to:
Distinction between expressions and equations and the difference between simplifying and
solving (TxCCRS)
Tools (e.g., algebraic properties)
Transformation of equations and inequalities using properties of equality and/or properties of
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Algebra II
TITLE : Unit 03: Linear Functions, Equations, and Inequalities
SUGGESTED DURATION : 13 days
TEKS#
TEKS
SE#
UNIT LEVEL SPECIFICITY
inequalities
Note:
In Algebra 1, students simplify expressions, apply rules of exponents, and solve linear and
quadratic equations.
TxCCRS Note:
II. Algebraic Reasoning A1 – Explain and differentiate between expressions and equations using words
such as “solve,” “evaluate,” and “simplify.”
II. Algebraic Reasoning B1 – Recognize and use algebraic (field) properties, concepts, procedures,
and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational
expressions).
II. Algebraic Reasoning C1 – Recognize and use algebraic (field) properties, concepts, procedures,
and algorithms to solve equations, inequalities, and systems of linear equations.
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),
Connect, Identify, Sketch
exponential (f (x) = ax), and logarithmic (f (x) =
logax) functions, absolute value of x (f (x) = |x|),
PARENT FUNCTIONS AND THEIR GRAPHS
square root of x (f (x) = √x), and reciprocal of x (f
(x) = 1/x).
Supporting Standard
Including, but not limited to:
General forms of parent functions (including equation and function notation)
Linear: f(x) = x
Representations of parent functions with and without technology
Tables
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Algebra II
TITLE : Unit 03: Linear Functions, Equations, and Inequalities
SUGGESTED DURATION : 13 days
TEKS#
TEKS
SE#
UNIT LEVEL SPECIFICITY
Graphs
Verbal descriptions
Algebraic generalizations (including equation and function notation)
Connections between representations
Characteristics of parent functions with and without technology
Ex: domain and range, rate of change, increasing/decreasing, intercepts, end behavior,
etc.
Connections between transformed functions to the respective parent function with and without
technology
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
Including, but not limited to:
General forms of parent functions (including equation and function notation)
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Algebra II
TITLE : Unit 03: Linear Functions, Equations, and Inequalities
SUGGESTED DURATION : 13 days
TEKS#
SE#
TEKS
UNIT LEVEL SPECIFICITY
Linear: f(x) = x
Representations of parent functions with and without technology
Graphs
Verbal descriptions
Algebraic generalizations (including equation and function notation)
Changes in parameters a, h, and k on the graph of the parent function: y = a(x - h) + k
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
Note:
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page 15 of 17 INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 03: Linear Functions, Equations, and Inequalities
SUGGESTED DURATION : 13 days
TEKS#
SE#
TEKS
UNIT LEVEL SPECIFICITY
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.
ELPS#
SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS.
The English Language Proficiency Standards (ELPS), as required by 19 Texas Administrative Code, Chapter 74, Subchapter A, §74.4, outline English language
proficiency 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.
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Algebra II
TITLE : Unit 03: Linear Functions, Equations, and Inequalities
SUGGESTED DURATION : 13 days
Last Updated 07/15/2014
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