PreCalculus Grades 1012 Page of
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Units: When plan is complete, right click on the first entry to refresh the table of contents. Unit 2  Graphs, Functions and Models Unit 3  Polynomial and Rational Functions Unit 4  Exponential and Logarithmic Functions Unit 5  Trigonometric Functions Unit 6  Analytic Trigonometry 
Unit Title  Unit 1  Algebra Readiness  Timeframe  3 weeks 
Unit Summary  In order to progress in the study of PreCalculus, students must have an understanding of the underlying algebraic topics such as linear functions, factoring, solving equations and solving inequalities.  
Learning Targets  
Essential Questions 
 
Enduring Understandings  Students will understand:
 
Know  By the end of this unit, students will know
 
Do  By the end of this unit, students will be able to
 
Evidence of Learning  
Formative 
 
Summative/ Benchmark 
 
Alternative Assessments 
 
Learning Activities  
 
Materials / Equipment / Resources  
Core Instructional Materials and Texts  Classroom Notes Khan Academy Flipped Math  
Equipment  TI83 Calculator Chromebook  Desmos Document Camera LCD Projector  
Supplemental Resources  Desmos Faceing Math Teachers Pay Teachers 
Standards  
Content Statement  Indicator  
Create equations that describe numbers or relationships.  Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.  
Represent and solve equations and inequalities graphically.  CCSS.MATH.CONTENT.HSA.REI.D.11 Explain why the xcoordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions  
Understand the relationship between zeros and factors of polynomials  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). 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.  
Use polynomial identities to solve problems.  Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2  y2)2 + (2xy)2 can be used to generate Pythagorean triples.  
Analyze functions using different representations.  CCSS.MATH.CONTENT.HSF.IF.C.7.C Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.  
Use complex numbers in polynomial identities and equations.  CCSS.MATH.CONTENT.HSN.CN.C.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.  
21st Century Skills and Themes  
Interdisciplinary Connections  Career Ready Practices  9.2 Career Awareness, Exploration, and Preparation 

 By the end of 12th grade,

Technology Standards  8.1  
912th Grade  
A. Technology Operations and Concepts: Students demonstrate a sound understanding of technology concepts, systems and operations.  
 8.1.12.A.2 Produce and edit a multipage 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.  
 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.  
 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.  
 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.  
 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 
 504s 

ELLs 
 G/T 

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 “Reallife” 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 
 
Enduring Understandings  Students will understand:
 
Know  By the end of this unit, students will know
 
Do  By the end of this unit, students will be able to
 
Evidence of Learning  
Formative 
 
Summative/ Benchmark 
 
Alternative Assessments 
 
Learning Activities  
 
Materials / Equipment / Resources  
Core Instructional Materials and Texts  Classroom Notes Khan Academy Flipped Math  
Equipment  TI83 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.  Understand 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  For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. 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.  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.B Graph square root, cube root, and piecewisedefined functions, including step functions and absolute value functions.  
Build new functions from existing functions.  Identify 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.  Rewrite 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 

 By the end of 12th grade,

Technology Standards  8.1  
912th Grade  
A. Technology Operations and Concepts: Students demonstrate a sound understanding of technology concepts, systems and operations.  
 8.1.12.A.2 Produce and edit a multipage 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.  
 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.  
 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.  
 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.  
 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 
 504s 

ELLs 
 G/T 

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 
 
Enduring Understandings  Students will understand:
 
Know  By the end of this unit, students will know
 
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 
 
Summative/ Benchmark 
 
Alternative Assessments 
 
Learning Activities  
 
Materials / Equipment / Resources  
Core Instructional Materials and Texts  Classroom Notes Khan Academy Flipped Math  
Equipment  TI83 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.  (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x  2i). (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.  
Understand the relationship between zeros and factors of polynomials.  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). 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.  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). 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.  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  Career Ready Practices  9.2 Career Awareness, Exploration, and Preparation 

 By the end of 12th grade,

Technology Standards  8.1  
912th Grade  
A. Technology Operations and Concepts: Students demonstrate a sound understanding of technology concepts, systems and operations.  
 8.1.12.A.2 Produce and edit a multipage 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.  
 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.  
 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.  
 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.  
 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 
 504s 

ELLs 
 G/T 

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 
 
Enduring Understandings  Students will understand:
 
Know  By the end of this unit, students will know
 
Do  By the end of this unit, students will be able to
 
Evidence of Learning  
Formative 
 
Summative/ Benchmark 
 
Alternative Assessments 
 
Learning Activities  
 
Materials / Equipment / Resources  
Core Instructional Materials and Texts  Classroom Notes Khan Academy Flipped Math  
Equipment  TI83 Calculator Chromebook  Desmos Document Camera LCD Projector  
Supplemental Resources  Desmos Faceing Math Teachers Pay Teachers 
Standards  
Content Statement  Indicator  
Analyze functions using different representations.  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.  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  Career Ready Practices  9.2 Career Awareness, Exploration, and Preparation 

 By the end of 12th grade,

Technology Standards  8.1  
912th Grade  
A. Technology Operations and Concepts: Students demonstrate a sound understanding of technology concepts, systems and operations.  
 8.1.12.A.2 Produce and edit a multipage 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.  
 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.  
 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.  
 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.  
 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 
 504s 

ELLs 
 G/T 

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 
 
Enduring Understandings  Students will understand:
 
Know  By the end of this unit, students will know
 
Do  By the end of this unit, students will be able to
 
Evidence of Learning  
Formative 
 
Summative/ Benchmark 
 
Alternative Assessments 
 
Learning Activities  
 
Materials / Equipment / Resources  
Core Instructional Materials and Texts  Classroom Notes Khan Academy Flipped Math  
Equipment  TI83 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.  Solve quadratic equations in one variable.  
Extend the domain of trigonometric functions using the unit circle.  Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. 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.  Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.* (+) 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.  Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*  
21st Century Skills and Themes  
Interdisciplinary Connections  Career Ready Practices  9.2 Career Awareness, Exploration, and Preparation 

 By the end of 12th grade,

Technology Standards  8.1  
912th Grade  
A. Technology Operations and Concepts: Students demonstrate a sound understanding of technology concepts, systems and operations.  
 8.1.12.A.2 Produce and edit a multipage 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.  
 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.  
 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.  
 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.  
 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 
 504s 

ELLs 
 G/T 

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 
 
Enduring Understandings  Students will understand:
 
Know  By the end of this unit, students will know
 
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 
 
Summative/ Benchmark 
 
Alternative Assessments 
 
Learning Activities  
 
Materials / Equipment / Resources  
Core Instructional Materials and Texts  Classroom Notes Khan Academy Flipped Math  
Equipment  TI83 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  Solve quadratic equations in one variable.  
Extend the domain of trigonometric functions using the unit circle.  Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. 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.  Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.* (+) 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.  Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*  
21st Century Skills and Themes  
Interdisciplinary Connections  Career Ready Practices  9.2 Career Awareness, Exploration, and Preparation 

 By the end of 12th grade,

Technology Standards  8.1  
912th Grade  
A. Technology Operations and Concepts: Students demonstrate a sound understanding of technology concepts, systems and operations.  
 8.1.12.A.2 Produce and edit a multipage 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.  
 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.  
 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.  
 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.  
 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 
 504s 

ELLs 
 G/T 
