UBD in a Nutshell https://www.dropbox.com/s/of71ubkbighhdij/UbD-in-a-Nutshell.pdf?dl=0

UBD QuickView - with example documents https://www.dropbox.com/s/wjahcqi1uv7b9u8/UbDQuikvue1005.pdf?dl=0

 Units: When plan is complete, right click on the first entry to refresh the table of contents. Unit 1 - Algebra ReadinessUnit 2 - Graphs, Functions and ModelsUnit 3 - Polynomial and Rational FunctionsUnit 4 - Exponential and Logarithmic FunctionsUnit 5 - Trigonometric FunctionsUnit 6 - Analytic Trigonometry

Unit Title

## Unit 1 - Algebra Readiness

Timeframe

3 weeks

Unit Summary

In order to progress in the study of Pre-Calculus, students must have an understanding of the underlying algebraic topics such as linear functions, factoring, solving equations and solving inequalities.

Learning Targets

Essential Questions

• How is mathematics used to measure, model and calculate change?
• How are polynomials factoried?
• What is the relationship between factoring and solving polynomial equations?
• How are exponents and radicals related?

Enduring Understandings

Students will understand:

• The relationship between the algebraic and graphical representations of a function.
• How to solve and represent the solution sets of algebraic equations and inequalities.

Know

By the end of this unit, students will know

• How to perform addition, subtraction, multiplication and division of polynomials
• How to factor polynomials
• How to solve and graph equations and inequalities
• How to simplify expressions with exponents and radicals

Do

By the end of this unit, students will be able to

• Perform operations on polynomials
• Factor polynomials
• Verbally, Algebraically, Graphically and Numerically represent equations and inequalities
• Perform operations with radicals and exponents

Evidence of Learning

Formative

• Person Puzzle: Eva Peron; Adding and Subtracting Polynomials
• Person Puzzle: Irena Sendler; Multiplying Monomials
• Person Puzzle: Stephanie Spielman; Multiplying Polynomials
• Person Puzzle: Wangari Maathai; Dividing Monomials and Negative Exponents
• Person Puzzle: Jaime Escalante; Factoring with a Greatest Common Factor
• Person Puzzle: Mother Theresa; Simple Factoring
• Person Puzzle: Muhammed Yunus; Advanced Factoring
• Person Puzzle: Grace Hopper; Quadratic Formula
• Person Puzzle: Maya Lin; Simplifying Rational Expressions
• Person Puzzle: Somaly Mam; Simplifying Radicals
• Person Puzzle: Hiroshi Yamauchi; Adding and Subtracting Rational Expressions
• Person Puzzle: Betty Mae Tiger Jumper; Multiplying and Dividing Rational Expressions
• Person Puzzle: Malala Yousafzai; Multiplying Radicals
• Person Puzzle: Desmond  Tutu; Multiplying Binomials with Radicals
• Person Puzzle: Ellen Degeneres; Rationalizing Denominators
• Person Puzzle: Hans Rosling; Solving Rational Equations

Summative/ Benchmark

• Unit Assessment: Monomial Boot Camp
• Unit Assessment: The relationship between the algebraic and graphical representations of a function.
• Unit Assessment:  How to solve and represent the solution sets of algebraic equations and inequalities

Alternative Assessments

• Mystery Adventure AND CSI Whodunnit: Multiplying Polynomials
• Mystery Adventure AND CSI Whodunnit: Dividing Monomials and Negative Exponents
• Mystery Adventure AND CSI Whodunnit: Factoring with GCF
• Mystery Adventure AND CSI Whodunnit: Simple Factoring
• Mystery Adventure AND CSI Whodunnit: Multiplying and Dividing Rational Expressions
• Mystery Adventure AND CSI Whodunnit: Rationalizing Denominators
• Mystery Adventure AND CSI Whodunnit: Solving Rational Equations

Learning Activities

• Group completion of graphic organizers involving equations, polynomials and inequalities
• Small group explorations involving equations, polynomials and inequalities
• Independent practice on algebra skills
• Using Kahoot to perform addition, subtraction, multiplication and division of polynomials
• Using Flipped Math to factor polynomials
• Graph equations and inequalities using Desmos
• Faceing Math Absolute Value and Inequalities
• Skill Practice Journey Game
• Writing Equations of Lines Jumble

Materials / Equipment / Resources

Core Instructional

Materials and Texts

Classroom Notes

Flipped Math

Equipment

TI-83 Calculator

Chromebook - Desmos

Document Camera

LCD Projector

Supplemental Resources

Desmos

Faceing Math

Teachers Pay Teachers

 Modifications/Accommodations IEPs preferential seating.extended time on tests and assignments.reduced homework or classwork.verbal, visual, or technology aids.modified textbooks or audio-video materials.behavior management support.adjusted class schedules or grading.verbal testing 504s preferential seating.extended time on tests and assignments.reduced homework or classwork.verbal, visual, or technology aids.modified textbooks or audio-video materials.behavior management support.adjusted class schedules or grading.verbal testing ELLs modeling and using gestures to aid in understanding.simplify instructions G/T provide learning centers where students are in charge of their learning.

Unit Title

## Unit 2 - Graphs, Functions and Models

Timeframe

5 weeks

Unit Summary

The concept of a function and function notation are central to modern mathematics and their applications.  This unit will develop skills in constructing and interpreting graphs of functions.  A “Real-life” situation in which one numerical quantity  depends on another is used to better understand the basic concepts of a function.  Graphs of many functions will be used to demonstrate characteristics of a function.

Learning Targets

Essential Questions

• How is mathematics used to measure, model and calculate change?
• How are functions and their graphs related?
• How can technology be used to investigate properties of families of functions and their graphs?
• What do effective problem solvers do, and what do they do when they get stuck?
•  How is mathematics used to quantify and compare situations, events and phenomena?

Enduring Understandings

Students will understand:

• Graphs and equations are alternative ways for depicting and analyzing patterns of linear and nonlinear change.
• Mathematical models can be used to describe physical relationships; these are often nonlinear.
• How different families of functions can be used to model real world situations.

Know

By the end of this unit, students will know

• How to graph a function with and without a graphing utility
• How to determine the domain and range of a function
• How to interpret the rate of change over an interval
• How to identify the characteristics and shape of various graphs
• How to graph the transformation of a function
• How to determine several ways in which given functions can be used to create new functions
• How to create functions based on a verbal description of real world situations

Do

By the end of this unit, students will be able to

• Determine if a relationship is a function
• Determine the inverse of a function and find the domain
• Determine if a function is 1 to 1
• Determine whether a function is odd, even or neither both algebraically and graphically
• Graph linear, absolute value, quadratic, square root, cube root, exponential, logarithmic and reciprocal functions with and without technology
• Identify the key characteristics of functions such as domain, range, maxima, minima, increasing intervals and decreasing intervals
• Analyze piece-wise functions
• Identify the vertex, intercepts, and axis of symmetry of a quadratic function
• Solve and model real-world problems using functions
• Transform functions
• Perform operations on functions

Evidence of Learning

Formative

• Person Puzzle: Serena Williams;  Identifying the Transformation
• Person Puzzle: Maria Montessori; Family of Functions with Transformations
• Working with Inverse Functions from a Table
• Analyzing Functions from Graphs and Tables group activity
• Write 5 higher order thinking questions and answer two.
• Tell me three things you know well and two things that confuse you.
• Play Kahoot.

Summative/ Benchmark

• Faceing Math: Operations with Functions
• Unit Assessment: Graph equations to depict and analyze patterns of linear and nonlinear change.
• Unit Assessment: Transforming and performing operations on functions.

Alternative Assessments

• Domain and Range Maze
• Transformations of Functions Task Cards
• Superhero Transformations Game
• Transformation of Functions Task Cards
• Writing Equations for Piecewise Functions Scavenger Hunt
• Model relationships to describe non-linear relationships.
• Make a visual representations of the different families of functions.
• Match, Pair and Share

Learning Activities

• Group completion of graphic organizers involving functions
• Small group explorations involving patterns of linear and nonlinear change
• Independent practice involving various types of functions
• Symmetry Jumble and Graphic Organizer
• Using Desmos to create functions from a real-world scenario
• Using Flipped Math to compare various types of functions
• Graph piece-wise functions using Desmos
• Identifying characteristics of functions through Khan Academy.

Materials / Equipment / Resources

Core Instructional

Materials and Texts

Classroom Notes

Flipped Math

Equipment

TI-83 Calculator

Chromebook - Desmos

Document Camera

LCD Projector

Supplemental Resources

Desmos

Faceing Math

Teachers Pay Teachers

 Standards Content Statement Indicator Understand the concept of a function and use function notation. CCSS.MATH.CONTENT.HSF.IF.A.1Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of fcorresponding to the input x. The graph of f is the graph of the equation y = f(x). Interpret functions that arise in applications in terms of the context CCSS.MATH.CONTENT.HSF.IF.B.4For 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity Analyze functions using different representations. CCSS.MATH.CONTENT.HSF.IF.C.7Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.CCSS.MATH.CONTENT.HSF.IF.C.7.BGraph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Build new functions from existing functions. CCSS.MATH.CONTENT.HSF.BF.B.3Identify the effect on the graph of replacing f(x) by f(x) + k, k 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 illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Rewrite rational expressions. CCSS.MATH.CONTENT.HSA.APR.D.6Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. 21st Century Skills and Themes Interdisciplinary Connections Career Ready Practices 9.2 Career Awareness, Exploration, and Preparation Language Arts through open-ended questions and self-reflections.  Business through the use of excel and finance.Science through physics problems. CRP1. Act as a responsible and contributing citizen and employee.CRP2. Apply appropriate academic and technical skills.CRP4.Communicate clearly and effectively and with reason.CRP6.Demonstrate creativity and innovation.CRP8.Utilize critical thinking to make sense of problems and persevere in solving them.CRP9.Model integrity, ethical leadership and effective management.CRP11. Use technology to enhance productivity.CRP12.Work productively in teams while using cultural global competence. By the end of 12th grade, 9.2.12.C.3 Identify transferable career skills and design alternate career plans. 9.2.12.C.4 Analyze how economic conditions and societal changes influence employment trends and Technology Standards - 8.1 9-12th Grade A. Technology Operations and Concepts: Students demonstrate a sound understanding of technology concepts, systems and operations. Select and use applications effectively and productively. 8.1.12.A.2 Produce and edit a multi-page digital document for a commercial or professional audience and present it to peers and/or professionals in that related area for review. 8.1.12.A.4 Construct a spreadsheet workbook with multiple worksheets, rename tabs to reflect the data on the worksheet, and use mathematical or logical functions, charts and data from all worksheets to convey the results. 8.1.12.A.5 Create a report from a relational database consisting of at least two tables and describe the process, and explain the report results. B. Creativity and Innovation: Students demonstrate creative thinking, construct knowledge and develop innovative products and process using technology. Apply existing knowledge to generate new ideas, products, or processes. Create original works as a means of personal or group expression. 8.1.12.B.2 Apply previous content knowledge by creating and piloting a digital learning game or tutorial. C. Communication and Collaboration: Students use digital media and environments to communicate and work collaboratively, including at a distance, to support individual learning and contribute to the learning of others. Interact, collaborate, and publish with peers, experts, or others by employing a variety of digital environments and media.Communicate information and ideas to multiple audiences using a variety of media and formats.Develop cultural understanding and global awareness by engaging with learners of other cultures.Contribute to project teams to produce original works or solve problems. 8.1.12.C.1 Develop an innovative solution to a real world problem or issue in collaboration with peers and experts, and present ideas for feedback through social media or in an online community. D. Digital Citizenship: Students understand human, cultural, and societal issues related to technology and practice legal and ethical behavior. Advocate and practice safe, legal, and responsible use of information and technology. 8.1.12.D.1 Demonstrate appropriate application of copyright, fair use and/or Creative Commons to an original work. F: Critical thinking, problem solving, and decision making: Students use critical thinking skills to plan and conduct research, manage projects, solve problems, and make informed decisions using appropriate digital tools and resources. Identify and define authentic problems and significant questions for investigation.Plan and manage activities to develop a solution or complete a project.Collect and analyze data to identify solutions and/or make informed decisions.Use multiple processes and diverse perspectives to explore alternative solutions. 8.1.12.F.1 Evaluate the strengths and limitations of emerging technologies and their impact on educational, career, personal and or social needs.

 Modifications/Accommodations IEPs preferential seating.extended time on tests and assignments.reduced homework or classwork.verbal, visual, or technology aids.modified textbooks or audio-video materials.behavior management support.adjusted class schedules or grading.verbal testing 504s preferential seating.extended time on tests and assignments.reduced homework or classwork.verbal, visual, or technology aids.modified textbooks or audio-video materials.behavior management support.adjusted class schedules or grading.verbal testing ELLs modeling and using gestures to aid in understanding.simplify instructions G/T provide learning centers where students are in charge of their learning.

Unit Title

## Unit 3 - Polynomial and Rational Functions

Timeframe

6 weeks

Unit Summary

Students will be able to examine different polynomial and rational functions that arise naturally in many applications.  More specifically, students will consider the degree, multiplicity, zeros, end behavior, and asymptotes when determining how a polynomial or rational function behaves.

Learning Targets

Essential Questions

• How do the characteristics of graphs relate to their corresponding equations?
• What are common characteristics of polynomial?
• What is the best method for graphing different polynomial and rational functions?

Enduring Understandings

Students will understand:

• Functions can be written in several ways
• Families of functions have common characteristics

Know

By the end of this unit, students will know

• Characteristics of polynomial and rational functions
• The relationship between zeros and factors of polynomials
• How to perform operations with complex numbers
• How to apply the Remainder and Factor Theorems
• How to find the degree, multiplicity, zeros, end behavior and asymptotes for polynomial and rational functions
• How to use the characteristics of polynomial and rational functions to create its’ graph
• How to use polynomial and rational functions to model real-world scenarios.
• How to identify common characteristics of families of functions

Do

By the end of this unit, students will be able to

• Examine graphs to determine end behavior
• Identify polynomial functions and find the degree given both factored and standard form
• Identify the zeros of a polynomial function and its multiplicity
• Investigate the role of multiplicity with regards to the graph of a polynomial function
• Investigate the role of the degree of a polynomial with regards to the number of turns a graph takes
• Factor polynomials of varying degrees
• Divide polynomials using long and synthetic division
• Apply the remainder and factor theorems
• Find the real and complex zeros of a polynomial
• Apply the conjugate pair theorem to help write a polynomial in standard form and find the zeros
• Form a polynomial with specified zeros both real and complex
• Solve polynomial equations for both real and complex zeros
• Find the domain, range, and asymptotes for polynomial and rational functions and use the information to create its graph
• Determine common characteristics of families of functions
• Analyze real world scenarios using polynomial and rational functions

Evidence of Learning

Formative

• Factoring Maze
• Person Puzzle: Bob Marley; Graphing Quadratic Functions
• Career: Football Scout - STEMersion with Graphing Quadratic Equations
• Career: Marketing Director - STEMersion with Quadratic Functions
• Working With Complex Numbers Jumble
• Person Puzzle: Alicia Alonso; Introduction to Imaginary Numbers
• Person Puzzle: Yuri Gagarin; Operations with Complex Numbers
• Person Puzzle: Catalina Escobar; Rationalizing Imaginary Denominators
• Career: Anesthesiologist - Solving and Graphing Rational Functions
• Using Demos to find the real zeros
• Describe the common characteristics of polynomials and rational functions.
• Explain to a friend how to transform a polynomial.
• Complex Numbers Maze
• Rational Functions Stations

Summative/ Benchmark

• Faceing Math: Graphs of Quadratics
• Unit Assessment: Graphing, solving, and transforming polynomials.
• Unit Assessment:  Solving polynomial and rational functions.

Alternative Assessments

• CSI Whodunnit: Factoring and Quadratic Functions
• CSI Whodunnit: Graphing Quadratic Functions
• Career Exploration: Marketing Maverick
• The Remainder and Factor Theorem Partnership
• CSI Whodunnit: Polynomial Functions
• Angry Penguins Adventure with Functions
• CSI Whodunnit: Add & Subtract Rational Expressions
• CSI Whodunnit: Multiply & Divide Rational Expressions
• CSI Whodunnit: Solving  Rational Expressions
• Journey Game: Rational Functions
• Birthday Polynomial
• Desmos Activities

Learning Activities

• Group completion of graphic organizers involving polynomial and rational functions
• Small group explorations involving polynomial and rational functions
• Independent practice with polynomial and rational functions
• Finding the Vertex of a Parabola Jumble
• Super Secret Number Puzzle: Synthetic Division
• Using Kahoot to reinforce the Remainder and Factor Theorems
• Color My Math: Working with Complex Numbers
• Using Flipped Math to show the difference between real and complex zeros
• Graphing Rational Equations Card Sort
• Using Desmos to compare graphs of families of functions
• Using Khan Academy to investigate the role of degree and multiplicity in a function

Materials / Equipment / Resources

Core Instructional

Materials and Texts

Classroom Notes

Flipped Math

Equipment

TI-83 Calculator

Chromebook - Desmos

Document Camera

LCD Projector

Supplemental Resources

Desmos

Faceing Math

Teachers Pay Teachers

Standards

Content Statement

Indicator

#### Use complex numbers in polynomial identities and equations.

CCSS.MATH.CONTENT.HSN.CN.C.8

(+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x - 2i).

CCSS.MATH.CONTENT.HSN.CN.C.9

(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

#### Understand the relationship between zeros and factors of polynomials.

CCSS.MATH.CONTENT.HSA.APR.B.2

Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).

CCSS.MATH.CONTENT.HSA.APR.B.3

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

#### Rewrite rational expressions.

CCSS.MATH.CONTENT.HSA.APR.B.2

Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).

CCSS.MATH.CONTENT.HSA.APR.B.3

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

#### Analyze functions using different representations.

CCSS.MATH.CONTENT.HSF.IF.C.7

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*

CCSS.MATH.CONTENT.HSF.IF.C.7.C

Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

21st Century Skills and Themes

Interdisciplinary Connections

9.2 Career Awareness, Exploration, and Preparation

• Language Arts through open-ended questions and self-reflections.
• Business through the use of excel and finance.
• Science through physics problems.

• CRP1. Act as a responsible and contributing citizen and employee.
• CRP2. Apply appropriate academic and technical skills.
• CRP4.Communicate clearly and effectively and with reason.
• CRP6.Demonstrate creativity and innovation.
• CRP8.Utilize critical thinking to make sense of problems and persevere in solving them.
• CRP9.Model integrity, ethical leadership and effective management.
• CRP11. Use technology to enhance productivity.
• CRP12.Work productively in teams while using cultural global competence.

By the end of 12th grade,

• 9.2.12.C.3 Identify transferable career skills and design alternate career plans.
• 9.2.12.C.4 Analyze how economic conditions and societal changes influence employment trends and

Technology Standards - 8.1

A. Technology Operations and Concepts: Students demonstrate a sound understanding of technology concepts, systems and operations.

• Select and use applications effectively and productively.

8.1.12.A.2 Produce and edit a multi-page digital document for a commercial or professional audience and present it to peers and/or professionals in that related area for review.

8.1.12.A.4 Construct a spreadsheet workbook with multiple worksheets, rename tabs to reflect the data on the worksheet, and use mathematical or logical functions, charts and data from all worksheets to convey the results.

8.1.12.A.5 Create a report from a relational database consisting of at least two tables and describe the process, and explain the report results.

B. Creativity and Innovation: Students demonstrate creative thinking, construct knowledge and develop innovative products and process using technology.

• Apply existing knowledge to generate new ideas, products, or processes.
• Create original works as a means of personal or group expression.

8.1.12.B.2 Apply previous content knowledge by creating and piloting a digital learning game or tutorial.

C. Communication and Collaboration: Students use digital media and environments to communicate and work collaboratively, including at a distance, to support individual learning and contribute to the learning of others.

• Interact, collaborate, and publish with peers, experts, or others by employing a variety of digital environments and media.
• Communicate information and ideas to multiple audiences using a variety of media and formats.
• Develop cultural understanding and global awareness by engaging with learners of other cultures.
• Contribute to project teams to produce original works or solve problems.

8.1.12.C.1 Develop an innovative solution to a real world problem or issue in collaboration with peers and experts, and present ideas for feedback through social media or in an online community.

D. Digital Citizenship: Students understand human, cultural, and societal issues related to technology and practice legal and ethical behavior.

• Advocate and practice safe, legal, and responsible use of information and technology.

8.1.12.D.1 Demonstrate appropriate application of copyright, fair use and/or Creative Commons to an original work.

F: Critical thinking, problem solving, and decision making: Students use critical thinking skills to plan and conduct research, manage projects, solve problems, and make informed decisions using appropriate digital tools and resources.

• Identify and define authentic problems and significant questions for investigation.
• Plan and manage activities to develop a solution or complete a project.
• Collect and analyze data to identify solutions and/or make informed decisions.
• Use multiple processes and diverse perspectives to explore alternative solutions.

8.1.12.F.1 Evaluate the strengths and limitations of emerging technologies and their impact on educational, career, personal and or social needs.

 Modifications/Accommodations IEPs preferential seating.extended time on tests and assignments.reduced homework or classwork.verbal, visual, or technology aids.modified textbooks or audio-video materials.behavior management support.adjusted class schedules or grading.verbal testing 504s preferential seating.extended time on tests and assignments.reduced homework or classwork.verbal, visual, or technology aids.modified textbooks or audio-video materials.behavior management support.adjusted class schedules or grading.verbal testing ELLs modeling and using gestures to aid in understanding.simplify instructions G/T provide learning centers where students are in charge of their learning.

Unit Title

## Unit 4 - Exponential and Logarithmic Functions

Timeframe

6 weeks

Unit Summary

This unit focuses on identifying and graphing exponential and logarithmic functions.  It is important to recognize that these functions are inverses of each other and have many applications in our world such as banking, medicine, growth and decay.

Learning Targets

Essential Questions

• What real-world scenarios can be represented by exponential and logarithmic functions?
• How can the properties of logarithms be used to solve equations?
• How can condensing and expanding exponents and logarithms be helpful in solving equations?

Enduring Understandings

Students will understand:

• The characteristics of exponential and logarithmic functions and their representations are useful in solving real-world problems
• Exponential and Logarithmic functions are closely related
• The importance of using exponential and logarithmic models to interpret real phenomena

Know

By the end of this unit, students will know

• The relationship between exponential and logarithmic functions
• How to simplify both exponential and logarithmic functions
• How to graph both exponential and logarithmic functions
• How to solve exponential and logarithmic equations
• How to apply exponential and logarithmic functions to real world situations

Do

By the end of this unit, students will be able to

• Evaluate exponential functions
• Graph exponential functions and their transformations
• Define the number e
• Solve exponential equations
• Change exponential expressions into logarithmic expressions and vice versa
• Evaluate logarithmic functions
• Graph logarithmic functions and their transformations
• Determine the domain and range of exponential and logarithmic functions
• Solve logarithmic equations
• Expand and condense logarithmic expressions
• Use the properties of exponents and logarithms to solve equations
• Use exponential and logarithmic functions to describe real world scenarios including growth and decay

Evidence of Learning

Formative

• Person Puzzle: Jesse Williams; Converting Exponents and Logarithms
• Person Puzzle: Tracy Chapman; Simplifying Logarithms
• Person Puzzle: Tony Dungy; Solving Exponential Equations without Logarithms
• Person Puzzle: Sonia Sotomayor; Solving Exponential Equations with Logarithms
• Person Puzzle: Millard Fuller; Solving Logarithmic Equations
• Career: Seismologist - STEMersion with Logarithmic Equations
• Career: Financial Advisor - STEMersion with Exponential Functions
• Using Demos to graph exponential and logarithmic functions.
• Describe the common characteristics of exponential and logarithmic function.
• Explain to a friend how to transform exponential and/or logarithmic function.

Summative/ Benchmark

• Faceing Math: Exponential Functions
• Exponential and Logarithmic Equations Card Sort
• Unit Assessment: Use exponential and logarithmic functions to describe real world scenarios including growth and decay
• Unit Assessment:  Evaluate exponential and logarithmic functions.

Alternative Assessments

• CSI Whodunnit: Converting Between Exponents and Logarithms
• CSI Whodunnit: Simplifying Logarithms
• CSI Whodunnit: Exponential Equations with Logarithms
• CSI Whodunnit: Logarithmic Equations
• Color My Math: Solving Logarithmic and Exponential Equations
• Solving Exponential Equations Maze
• Solving Logarithmic Equations Maze
• Graph logarithmic functions and their transformations

Learning Activities

• Group completion of graphic organizers involving exponents and logarithms
• Small group explorations involving exponents and logarithms
• Independent practice on exponential and logarithmic functions
• Simplifying Logarithms Card Sort
• Exponential, Logistic, Logarithmic Lab Activity
• Using Kahoot to perform expanding and condensing logarithmic expressions
• Using Flipped Math to show the relationship between exponents and logarithms
• Graph exponential and logarithmic functions and their transformations using Desmos
• Solve exponential and logarithmic equations with Khan Academy.

Materials / Equipment / Resources

Core Instructional

Materials and Texts

Classroom Notes

Flipped Math

Equipment

TI-83 Calculator

Chromebook - Desmos

Document Camera

LCD Projector

Supplemental Resources

Desmos

Faceing Math

Teachers Pay Teachers

Standards

Content Statement

Indicator

#### Analyze functions using different representations.

CCSS.MATH.CONTENT.HSF.IF.C.7

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*

#### Analyze functions using different representations.

CCSS.MATH.CONTENT.HSF.IF.C.7.E

Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

#### Construct and compare linear, quadratic, and exponential models and solve problems.

CCSS.MATH.CONTENT.HSF.LE.A.4

For exponential models, express as a logarithm the solution to abct = d where a, c, and dare numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

21st Century Skills and Themes

Interdisciplinary Connections

9.2 Career Awareness, Exploration, and Preparation

• Language Arts through open-ended questions and self-reflections.
• Business through the use of excel and finance.
• Science through physics problems.

• CRP1. Act as a responsible and contributing citizen and employee.
• CRP2. Apply appropriate academic and technical skills.
• CRP4.Communicate clearly and effectively and with reason.
• CRP6.Demonstrate creativity and innovation.
• CRP8.Utilize critical thinking to make sense of problems and persevere in solving them.
• CRP9.Model integrity, ethical leadership and effective management.
• CRP11. Use technology to enhance productivity.
• CRP12.Work productively in teams while using cultural global competence.

By the end of 12th grade,

• 9.2.12.C.3 Identify transferable career skills and design alternate career plans.
• 9.2.12.C.4 Analyze how economic conditions and societal changes influence employment trends and

Technology Standards - 8.1

A. Technology Operations and Concepts: Students demonstrate a sound understanding of technology concepts, systems and operations.

• Select and use applications effectively and productively.

8.1.12.A.2 Produce and edit a multi-page digital document for a commercial or professional audience and present it to peers and/or professionals in that related area for review.

8.1.12.A.4 Construct a spreadsheet workbook with multiple worksheets, rename tabs to reflect the data on the worksheet, and use mathematical or logical functions, charts and data from all worksheets to convey the results.

8.1.12.A.5 Create a report from a relational database consisting of at least two tables and describe the process, and explain the report results.

B. Creativity and Innovation: Students demonstrate creative thinking, construct knowledge and develop innovative products and process using technology.

• Apply existing knowledge to generate new ideas, products, or processes.
• Create original works as a means of personal or group expression.

8.1.12.B.2 Apply previous content knowledge by creating and piloting a digital learning game or tutorial.

C. Communication and Collaboration: Students use digital media and environments to communicate and work collaboratively, including at a distance, to support individual learning and contribute to the learning of others.

• Interact, collaborate, and publish with peers, experts, or others by employing a variety of digital environments and media.
• Communicate information and ideas to multiple audiences using a variety of media and formats.
• Develop cultural understanding and global awareness by engaging with learners of other cultures.
• Contribute to project teams to produce original works or solve problems.

8.1.12.C.1 Develop an innovative solution to a real world problem or issue in collaboration with peers and experts, and present ideas for feedback through social media or in an online community.

D. Digital Citizenship: Students understand human, cultural, and societal issues related to technology and practice legal and ethical behavior.

• Advocate and practice safe, legal, and responsible use of information and technology.

8.1.12.D.1 Demonstrate appropriate application of copyright, fair use and/or Creative Commons to an original work.

F: Critical thinking, problem solving, and decision making: Students use critical thinking skills to plan and conduct research, manage projects, solve problems, and make informed decisions using appropriate digital tools and resources.

• Identify and define authentic problems and significant questions for investigation.
• Plan and manage activities to develop a solution or complete a project.
• Collect and analyze data to identify solutions and/or make informed decisions.
• Use multiple processes and diverse perspectives to explore alternative solutions.

8.1.12.F.1 Evaluate the strengths and limitations of emerging technologies and their impact on educational, career, personal and or social needs.

 Modifications/Accommodations IEPs preferential seating.extended time on tests and assignments.reduced homework or classwork.verbal, visual, or technology aids.modified textbooks or audio-video materials.behavior management support.adjusted class schedules or grading.verbal testing 504s preferential seating.extended time on tests and assignments.reduced homework or classwork.verbal, visual, or technology aids.modified textbooks or audio-video materials.behavior management support.adjusted class schedules or grading.verbal testing ELLs modeling and using gestures to aid in understanding.simplify instructions G/T provide learning centers where students are in charge of their learning.

Unit Title

## Unit 5 - Trigonometric Functions

Timeframe

12 weeks

Unit Summary

Students will be introduced to two ways to identify the six trigonometric functions; one using right triangles and one using the Unit Circle.  The real world applications that involve trigonometry are endless and include: height of a tree, length of a ramp, distance between cities, construction of a building or bridge, differences between radio and sound waves, etc.  The graphs of the trigonometric functions and their transformations will be investigated as well.

Learning Targets

Essential Questions

• How can the Unit Circle be used to evaluate and graph the six trigonometric functions?
• How does the equation of a trigonometric function affect its graph?
• How can we use trigonometric functions to describe physical relationships?
• How can we calculate exact values for angles that are not found using the Unit Circle?
• Why must a trigonometric function’s domain be restricted to construct its inverse?

Enduring Understandings

Students will understand:

• Trigonometric functions can be used to describe and quantify relationships
• The Unit Circle can be used to calculate the six trigonometric functions at a given point
• Having an understanding of simple identities can open doors to a whole new set of identities
• Identities can be used to calculate exact trigonometric values for any angle using known special triangles
• There exists distinct and definite connections between algebra and geometry
• There are great applications in the real world that require the knowledge of trigonometry

Know

By the end of this unit, students will know

• How to solve for the missing sides and angles in a right triangle
• The relationship between the six trigonometric functions and the Unit Circle
• The relationship between degrees and radians
• How to graph and calculate coterminal and reference angles
• How to graph the six trigonometric functions and their transformations
• How to determine the amplitude, period, domain and range for the six trigonometric functions and their transformations
• How to use trigonometry to solve real-life applications
• How to graph and solve inverse trigonometric functions

Do

By the end of this unit, students will be able to

• Use right triangle trigonometry to find missing parts of a right triangle including real-life applications
• Identify coterminal and reference angles using degrees and radians
• Create a Unit Circle using right triangle trigonometry
• Use the Unit Circle to evaluate all six trigonometric functions
• Evaluate trigonometric functions of any angle using reference angles
• Graph the sine and cosine from the Unit Circle
• Sketch the graphs of the tangent and cotangent functions
• Sketch the graphs of secant and cosecant functions using the sine and cosine graphs
• Describe important characteristics of the graphs of the six trigonometric functions including period, amplitude, domain and range
• Graph transformed trigonometric functions
• Discuss the restrictions for the domain on inverse trigonometric functions
• Calculate the exact values of the inverse trigonometric functions

Evidence of Learning

Formative

• Person Puzzle: Audra Lorde; Radian and Degree Measure
• Trigonometry Special Values Puzzle
• Faceing Math: Basic Trigonometry
• Trigonometric Functions Relay Race
• Faceing Math: More Trigonometry
• Sketch the graphs of the tangent and cotangent functions
• Sketch the graphs of secant and cosecant functions using the sine and cosine graphs
• Discuss the restrictions for the domain on inverse trigonometric functions

Summative/ Benchmark

• Unit Assessment: How to graph the six trigonometric functions and their transformations
• Unit Assessment:  Use the Unit Circle to evaluate all six trigonometric functions

Alternative Assessments

• CSI Whodunnit: Converting Radians and Degrees
• Angle Tangle Spider Web
• Trigonometric Ratios Coloring - Superman
• Right Triangle Trigonometry Scavenger Hunt
• Trigonometry Matching Square
• Graph Matching Bingo
• Trigonometric Art Project
• Match, Pair and Share
• Create a Unit Circle using right triangle trigonometry
• Use right triangle trigonometry to find missing parts of a right triangle including real-life applications

Learning Activities

• Group completion of graphic organizers involving trigonometric functions and the Unit Circle
• Small group explorations involving trigonometric functions and the Unit Circle
• Independent practice involving trigonometric functions and the Unit Circle
• Digital Interactive Radian to Degree Puzzle
• Right Triangle Coloring - Rooster
• Star Wars Jedi Academy: Applications of Trigonometry
• Trigonometry Convenient Values Flash Cards
• Trigonometry Jeopardy
• Digital Interactive Trigonometry Values Puzzle
• Create a Unit Circle out of garland
• Create a Unit Circle picture
• Create the sine and cosine graphs out of pasta
• Trigonometric Match-Up Activity
• Inverse Trigonometric Functions Scavenger Hunt
• Using Kahoot to find coterminal and reference angles
• Using Flipped Math to solve right triangle real-world scenarios
• Graph trigonometric functions and their translations using Desmos
• Solve inverse trigonometric functions with Khan Academy

Materials / Equipment / Resources

Core Instructional

Materials and Texts

Classroom Notes

Flipped Math

Equipment

TI-83 Calculator

Chromebook - Desmos

Document Camera

LCD Projector

Supplemental Resources

Desmos

Faceing Math

Teachers Pay Teachers

Standards

Content Statement

Indicator

#### Solve equations and inequalities in one variable.

CCSS.MATH.CONTENT.HSA.REI.B.4

Solve quadratic equations in one variable.

#### Extend the domain of trigonometric functions using the unit circle.

CCSS.MATH.CONTENT.HSF.TF.A.1

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

CCSS.MATH.CONTENT.HSF.TF.A.2

Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

#### Model periodic phenomena with trigonometric functions.

CCSS.MATH.CONTENT.HSF.TF.B.5

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.*

CCSS.MATH.CONTENT.HSF.TF.B.7

(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.*

#### Define trigonometric ratios and solve problems involving right triangles.

CCSS.MATH.CONTENT.HSG.SRT.C.8

Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*

21st Century Skills and Themes

Interdisciplinary Connections

9.2 Career Awareness, Exploration, and Preparation

• Language Arts through open-ended questions and self-reflections.
• Business through the use of excel and finance.
• Science through physics problems.

• CRP1. Act as a responsible and contributing citizen and employee.
• CRP2. Apply appropriate academic and technical skills.
• CRP4.Communicate clearly and effectively and with reason.
• CRP6.Demonstrate creativity and innovation.
• CRP8.Utilize critical thinking to make sense of problems and persevere in solving them.
• CRP9.Model integrity, ethical leadership and effective management.
• CRP11. Use technology to enhance productivity.
• CRP12.Work productively in teams while using cultural global competence.

By the end of 12th grade,

• 9.2.12.C.3 Identify transferable career skills and design alternate career plans.
• 9.2.12.C.4 Analyze how economic conditions and societal changes influence employment trends and

Technology Standards - 8.1

A. Technology Operations and Concepts: Students demonstrate a sound understanding of technology concepts, systems and operations.

• Select and use applications effectively and productively.

8.1.12.A.2 Produce and edit a multi-page digital document for a commercial or professional audience and present it to peers and/or professionals in that related area for review.

8.1.12.A.4 Construct a spreadsheet workbook with multiple worksheets, rename tabs to reflect the data on the worksheet, and use mathematical or logical functions, charts and data from all worksheets to convey the results.

8.1.12.A.5 Create a report from a relational database consisting of at least two tables and describe the process, and explain the report results.

B. Creativity and Innovation: Students demonstrate creative thinking, construct knowledge and develop innovative products and process using technology.

• Apply existing knowledge to generate new ideas, products, or processes.
• Create original works as a means of personal or group expression.

8.1.12.B.2 Apply previous content knowledge by creating and piloting a digital learning game or tutorial.

C. Communication and Collaboration: Students use digital media and environments to communicate and work collaboratively, including at a distance, to support individual learning and contribute to the learning of others.

• Interact, collaborate, and publish with peers, experts, or others by employing a variety of digital environments and media.
• Communicate information and ideas to multiple audiences using a variety of media and formats.
• Develop cultural understanding and global awareness by engaging with learners of other cultures.
• Contribute to project teams to produce original works or solve problems.

8.1.12.C.1 Develop an innovative solution to a real world problem or issue in collaboration with peers and experts, and present ideas for feedback through social media or in an online community.

D. Digital Citizenship: Students understand human, cultural, and societal issues related to technology and practice legal and ethical behavior.

• Advocate and practice safe, legal, and responsible use of information and technology.

8.1.12.D.1 Demonstrate appropriate application of copyright, fair use and/or Creative Commons to an original work.

F: Critical thinking, problem solving, and decision making: Students use critical thinking skills to plan and conduct research, manage projects, solve problems, and make informed decisions using appropriate digital tools and resources.

• Identify and define authentic problems and significant questions for investigation.
• Plan and manage activities to develop a solution or complete a project.
• Collect and analyze data to identify solutions and/or make informed decisions.
• Use multiple processes and diverse perspectives to explore alternative solutions.

8.1.12.F.1 Evaluate the strengths and limitations of emerging technologies and their impact on educational, career, personal and or social needs.

 Modifications/Accommodations IEPs preferential seating.extended time on tests and assignments.reduced homework or classwork.verbal, visual, or technology aids.modified textbooks or audio-video materials.behavior management support.adjusted class schedules or grading.verbal testing 504s preferential seating.extended time on tests and assignments.reduced homework or classwork.verbal, visual, or technology aids.modified textbooks or audio-video materials.behavior management support.adjusted class schedules or grading.verbal testing ELLs modeling and using gestures to aid in understanding.simplify instructions G/T provide learning centers where students are in charge of their learning.

Unit Title

## Unit 6 - Analytic Trigonometry

Timeframe

8 weeks

Unit Summary

Solving trigonometric equations can be helpful when trying to find the measure of an angle or the area of a triangle.  Students will prove the Pythagorean Identities and use them to establish new identities.  These identities will help them to solve for angles and sides in both right and oblique triangles.

Learning Targets

Essential Questions

• What does it mean to solve a trigonometric equation?
• How are multiple solutions represented on a given domain?
• What does it mean to prove or verify a trigonometric expression?
• How can identities and algebraic methods be used to help solve trigonometric equations?
• Why does the Law of Sines have an ambiguous case?
• How can trigonometry be used to solve any type of triangle and find its area?

Enduring Understandings

Students will understand:

• Equivalent expressions can be written in infinite ways
• The solutions to a trigonometric equation can be represented algebraically and graphically
• Trigonometry can be used to solve oblique as well as right triangles
• Real-world phenomena can be modeled using trigonometric principles

Know

By the end of this unit, students will know

• How to use the trigonometric identities to simplify, verify and prove expressions
• How to solve trigonometric equations
• How to apply the sum, difference, double and half-angle formulas
• How to apply the Law of Sines and the Law of Cosines to solve for the sides and angles in oblique triangles
• How to find the area of triangles using trigonometry

Do

By the end of this unit, students will be able to

• Verify and prove trigonometric expressions using the identities
• Solve trigonometric equations using algebraic and graphic techniques
• Find the exact values of trigonometric functions using the sum and difference formulas
• Find the exact values of trigonometric functions using the double and half angle formulas
• Use the Law of Sines and the Law of Cosines to solve oblique triangles
• Find the area of triangles using the Sine function
• Use Heron’s Formula to find the area of a triangle

Evidence of Learning

Formative

• Trigonometric Identities Puzzle
• Faceing Math: Trigonometric Identities
• Solving Trigonometric Equations Task Cards
• Person Puzzle: Priscilla Chan; Sum and Difference Identities
• Person Puzzle: Nancy Wake; Double and Half Angle Identities
• Solving Trigonometric Equations Task Cards
• Career: Surveyor - STEMersion with Law of Sines
• Person Puzzle: Michael J. Fox; Law of Sines
• Person Puzzle: Marie Curie; Law of Cosines
• Area of Oblique Triangles Task Cards

Summative/ Benchmark

• Unit Assessment: Solving Trigonometric Equations using Identities
• Unit Assessment:  Solving and finding Area of Oblique Triangles

Alternative Assessments

• Fill in the blank: The Story of Joe Sine
• Trigonometric Identities Magic Square
• Color My Math: Solving trigonometric Equations
• CSI Whodunnit: Sum and Difference Identities
• CSI Whodunnit: Law of Sines
• Coloring: Law of Sines Bird
• CSI Whodunnit: Law of Cosines
• CSI Whodunnit: Multi-step Trigonometry
• Law of Sines and Cosines Maze
• Law of SInes and Cosines Coloring - Batman

Learning Activities

• Group completion of graphic organizers involving analytic trigonometry
• Small group explorations involving analytic trigonometry
• Independent practice involving analytic trigonometry
• Practicing Trigonometric Identities People Search
• Proving Trigonometric Identities Cut and Paste Activity
• Solving Trigonometric Equations - Peter Pan Activity
• Introduction to Solving Trigonometric Equations Using Desmos
• Using Kahoot to apply the sum, difference, double and half-angle formulas
• Using Flipped Math to discover the Law of Sines and the Law of Cosines formulas
• Find the area of triangles using Desmos
• Verify and prove trigonometric expressions with Khan Academy.

Materials / Equipment / Resources

Core Instructional

Materials and Texts

Classroom Notes

Flipped Math

Equipment

TI-83 Calculator

Chromebook - Desmos

Document Camera

LCD Projector

Supplemental Resources

Desmos

Faceing Math

Teachers Pay Teachers

Standards

Content Statement

Indicator

#### Solve equations and inequalities in one variable

CCSS.MATH.CONTENT.HSA.REI.B.4

Solve quadratic equations in one variable.

#### Extend the domain of trigonometric functions using the unit circle.

CCSS.MATH.CONTENT.HSF.TF.A.1

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

CCSS.MATH.CONTENT.HSF.TF.A.2

Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

#### Model periodic phenomena with trigonometric functions.

CCSS.MATH.CONTENT.HSF.TF.B.5

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.*

CCSS.MATH.CONTENT.HSF.TF.B.7

(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.*

#### Define trigonometric ratios and solve problems involving right triangles.

CCSS.MATH.CONTENT.HSG.SRT.C.8

Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*

21st Century Skills and Themes

Interdisciplinary Connections

9.2 Career Awareness, Exploration, and Preparation

• Language Arts through open-ended questions and self-reflections.
• Business through the use of excel and finance.
• Science through physics problems.

• CRP1. Act as a responsible and contributing citizen and employee.
• CRP2. Apply appropriate academic and technical skills.
• CRP4.Communicate clearly and effectively and with reason.
• CRP6.Demonstrate creativity and innovation.
• CRP8.Utilize critical thinking to make sense of problems and persevere in solving them.
• CRP9.Model integrity, ethical leadership and effective management.
• CRP11. Use technology to enhance productivity.
• CRP12.Work productively in teams while using cultural global competence.

By the end of 12th grade,

• 9.2.12.C.3 Identify transferable career skills and design alternate career plans.
• 9.2.12.C.4 Analyze how economic conditions and societal changes influence employment trends and

Technology Standards - 8.1

A. Technology Operations and Concepts: Students demonstrate a sound understanding of technology concepts, systems and operations.

• Select and use applications effectively and productively.

8.1.12.A.2 Produce and edit a multi-page digital document for a commercial or professional audience and present it to peers and/or professionals in that related area for review.

8.1.12.A.4 Construct a spreadsheet workbook with multiple worksheets, rename tabs to reflect the data on the worksheet, and use mathematical or logical functions, charts and data from all worksheets to convey the results.

8.1.12.A.5 Create a report from a relational database consisting of at least two tables and describe the process, and explain the report results.

B. Creativity and Innovation: Students demonstrate creative thinking, construct knowledge and develop innovative products and process using technology.

• Apply existing knowledge to generate new ideas, products, or processes.
• Create original works as a means of personal or group expression.

8.1.12.B.2 Apply previous content knowledge by creating and piloting a digital learning game or tutorial.

C. Communication and Collaboration: Students use digital media and environments to communicate and work collaboratively, including at a distance, to support individual learning and contribute to the learning of others.

• Interact, collaborate, and publish with peers, experts, or others by employing a variety of digital environments and media.
• Communicate information and ideas to multiple audiences using a variety of media and formats.
• Develop cultural understanding and global awareness by engaging with learners of other cultures.
• Contribute to project teams to produce original works or solve problems.

8.1.12.C.1 Develop an innovative solution to a real world problem or issue in collaboration with peers and experts, and present ideas for feedback through social media or in an online community.

D. Digital Citizenship: Students understand human, cultural, and societal issues related to technology and practice legal and ethical behavior.

• Advocate and practice safe, legal, and responsible use of information and technology.

8.1.12.D.1 Demonstrate appropriate application of copyright, fair use and/or Creative Commons to an original work.

F: Critical thinking, problem solving, and decision making: Students use critical thinking skills to plan and conduct research, manage projects, solve problems, and make informed decisions using appropriate digital tools and resources.

• Identify and define authentic problems and significant questions for investigation.
• Plan and manage activities to develop a solution or complete a project.
• Collect and analyze data to identify solutions and/or make informed decisions.
• Use multiple processes and diverse perspectives to explore alternative solutions.

8.1.12.F.1 Evaluate the strengths and limitations of emerging technologies and their impact on educational, career, personal and or social needs.

 Modifications/Accommodations IEPs preferential seating.extended time on tests and assignments.reduced homework or classwork.verbal, visual, or technology aids.modified textbooks or audio-video materials.behavior management support.adjusted class schedules or grading.verbal testing 504s preferential seating.extended time on tests and assignments.reduced homework or classwork.verbal, visual, or technology aids.modified textbooks or audio-video materials.behavior management support.adjusted class schedules or grading.verbal testing ELLs modeling and using gestures to aid in understanding.simplify instructions G/T provide learning centers where students are in charge of their learning.