Unit IV: Evolution s3

Pre-Calculus

Course Description and Philosophy

The Pre-Calculus course presented at the high school is intended to serve those students who have successfully completed the course Algebra 2. The purpose of the course is to continue the study of elementary functions including topics in algebra, geometry, and especially trigonometry. The course is geared at preparing the student for their first course in Calculus or other collegiate mathematics courses. In short, Pre-Calculus provides experiences that encourage and enable students to value mathematics, gain confidence in their own mathematical ability, become mathematical problem solvers, communicate mathematically, and reason mathematically. Further, students will use current technology such as the graphing calculator on a day-to-day basis. This course further reflects the needs of such capable students and recognizes the fact that they will spend their adult lives in a society increasingly dominated by technology and quantitative methods.

Teacher Resources: Illustrative Mathematics Content Standards: High School http://www.illustrativemathematics.org/standards/hs The Teaching Channel (Common Core Math Channel) https://www.teachingchannel.org/videos?page=1&categories=subjects_math,topics_common-core&load=1 The Mathematics Common Core Toolbox (http://ccsstoolbox.agilemind.com/resources_samples.html) has both sample scope and sequence documents as well grades 4-12 PARRC assessment tasks.

Text Reference: Brown, Richard G. Advanced Mathematics, 1992 Houghton Mifflin Company, Boston, MA / USA

Revised 2013

1 Unit I: Functions and Models Essential Question: Why are relations and functions represented in multiple ways?

Objectives: Students will be able to  Represent functions and relations numerically, graphically, and algebraically.  Describe and apply properties of functions and relations.  Perform a variety of operations and geometrical transformations on functions and relations.  Use numerical, algebraic, and graphical representations of functions and relations in order to solve real-world problems.

Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS/CC Represent a piece-wise Tests/Quizzes Text Lectures Graphing calculator 1.1.PC.8 function numerically, Homework Worksheets Discussion SmartBoard algebraically, and Class Participation Investigations Links to “Flipped graphically. Class Investigations Activities Classroom” resources: (e.g. Topic/Content Skills Assessment Resources Instructional Method Khan Tech Academy Infusion) NJCCCS/CC Identify and distinguish Tests/Quizzes Text Lectures Graphing calculator 1.2.PC.6 betweenDescribe the the graphs properties of HomeworkTests/Quizzes WorksheetsText DiscussionLectures SmartBoardGraphing calculator 1.2.PC.3 linear,of linear, quadratic, quadratic, power, ClassHomework Participation Worksheets InvestigationsDiscussion LinksSmartBoard to “Flipped Links to “Flipped polynomial,power, polynomial, rational, ClassClass Investigations Participation ActivitiesInvestigations Classroom” resources: (e.g. rational, exponential, Class Investigations Activities KhanClassroom” Academy resources:) (e.g. exponential, logarithmic, Khan Academy) trigonometric,logarithmic, and inversetrigonometric, trigonometric and functions.inverse trigonometric functions DescribeDescribe the the effect properties of Tests/QuizzesTests/Quizzes TextText LecturesLectures GraphingGraphing calculator calculator 1.3.PC.11.2.PC.4 singleof a piece-wise or multiple function. HomeworkHomework WorksheetsWorksheets DiscussionDiscussion SmartBoardSmartBoard Links to “Flipped transformations on the ClassClass Participation Participation InvestigationsInvestigations Links to “Flipped Classroom” resources: (e.g. Class Investigations Activities Classroom” resources: (e.g. graph of f(x), including Class Investigations Activities Khan Academy) a f(x), f⎛x−h⎞, f(x)+k, Khan Academy) f(ax), f x, and f⎛x⎞, f(| x|), |f(x)|

2 Describe the effect of Tests/Quizzes Text Lectures Graphing calculator 1.3.PC.5 transformations on a Homework Worksheets Discussion SmartBoard function with a restricted Class Participation Investigations Links to “Flipped domain. Class Investigations Activities Classroom” resources: (e.g. Khan Academy)

Interpret and solve Tests/Quizzes Text Lectures Graphing calculator 1.4.PC.9 problems involving Homework Worksheets Discussion SmartBoard piece-wise functions. Class Participation Investigations Links to “Flipped Class Investigations Activities Classroom” resources: (e.g. Khan Academy)

Make predictions using Tests/Quizzes Text Lectures Graphing calculator 1.4.PC.12 trigonometric or power Homework Worksheets Discussion SmartBoard mathematical models Class Participation Investigations Links to “Flipped given a set of data. Class Investigations Activities Classroom” resources: (e.g. Khan Academy)

Differentiated Learning Activities Strategic Learner: Break problems into distinct steps, verbalizing the reasoning process. Encourage the use of the graphing calculator. Students should refer to Khan tutorials for additional support. Advanced Learner: Pose journal problems. Discuss ill-posed problems. Students may create their own tutorials to help peers and to demonstrate their own mastery.

Ethical Decision Making/Character Education Discuss the history of the quintic solution including the issue of plagiarism regarding Ferrari, Cardano, and Tartaglia.

21st Century Skills Students use graphing calculators, computer algebra systems, or both, to assist in examining patterns in population growth or decline of a particular animal species using data from the U.S. Fish & Wildlife Service or another source. For example, students might study the changes in the population of Canada geese by building functions and examining graphs to answer problems regarding long-term trends. Students can share their findings with a school biology class by developing a lesson on the use of mathematics in examining biological developments.

Unit II: Trigonometric Functions Essential Question: How do trigonometric and circular functions model real- world problems and their solutions? 3

Objectives: Students will be able to  Represent functions and relations numerically, graphically, and algebraically.  Describe and apply properties of functions and relations.  Perform a variety of operations and geometrical transformations on functions and relations.  Use numerical, algebraic, and graphical representations of functions and relations in order to solve real-world problems.  Analyze two- and three-dimensional figures using tools and technology when appropriate.  Apply geometric properties and relationships to solve problems using tools and technology when appropriate

Topic/Content Skills Assessment Materials Instructional Method Tech Infusion NJCCCS/CC Determine the period, Tests/Quizzes Text Lectures Graphing calculator 1.1.PC.15 amplitude, phase shift, Homework Worksheets Discussion SmartBoard and/or vertical shift of a Class Participation Investigations Links to “Flipped Classroom” trigonometric function Class Investigations Activities resources: (e.g. Khan Academy) represented graphically or algebraically.

Unit 2 Topic/Content Skills Assessment Materials Instructional Method Tech Infusion NJCCCS/CC Describe the properties of Tests/Quizzes Text Lectures Graphing calculator 1.2.PC.3 linear, quadratic, power, Homework Worksheets Discussion SmartBoard polynomial, rational, Class Participation Investigations Links to “Flipped Classroom” exponential, logarithmic, Investigations Activities resources: (e.g. Khan Academy) trigonometric, and inverse trigonometric functions. Describe the inverse Tests/Quizzes Text Lectures Graphing calculator 1.2.PC.5 relationship between Homework Worksheets Discussion SmartBoard trigonometric and inverse Class Participation Investigations Links to “Flipped Classroom” trigonometric functions, Investigations Activities resources: (e.g. Khan Academy) numerically, algebraically, and graphically.

4 Identify and distinguish Tests/Quizzes Text Lectures Graphing calculator 1.2.PC.6 between the graphs of Homework Worksheets Discussion SmartBoard linear, quadratic, power, Class Participation Investigations Links to “Flipped Classroom” polynomial, rational, Investigations Activities resources: (e.g. Khan Academy) exponential, logarithmic, trigonometric, and inverse trigonometric functions Describe the effect of Tests/Quizzes Text Lectures Graphing calculator 1.3.PC.4 transformations on the Homework Worksheets Discussion SmartBoard graphs of trigonometric Class Participation Investigations Links to “Flipped Classroom” functions. Investigations Activities resources: (e.g. Khan Academy)

Interpret and solve Tests/Quizzes Text Lectures Graphing calculator 1.4.PC.10 problems involving Homework Worksheets Discussion SmartBoard trigonometric functions. Class Participation Investigations Links to “Flipped Classroom” Investigations Activities resources: (e.g. Khan Academy)

Unit 2 Topic/Content Skills Assessment Materials Instructional Method Tech Infusion NJCCCS/CC Make predictions using Tests/Quizzes Text Lectures Graphing calculator 1.4.PC.12 trigonometric or power Homework Worksheets Discussion SmartBoard mathematical models Class Participation Investigations Links to “Flipped Classroom” given a set of data.. Investigations Activities resources: (e.g. Khan Academy)

Describe and apply the Tests/Quizzes Text Lectures Graphing calculator 2.1.PC.4 relationship between the Homework Worksheets Discussion SmartBoard trigonometry of the right Class Participation Investigations Links to “Flipped Classroom” triangle and the unit Investigations Activities resources: (e.g. Khan Academy) circle.

5 Describe and apply the Tests/Quizzes Text Lectures Graphing calculator 2.1.PC.5 relationship between the Homework Worksheets Discussion SmartBoard radian measure of a Class Participation Investigations Links to “Flipped Classroom” central angle of a circle Investigations Activities resources: (e.g. Khan Academy) and its intercepted arc. Define and graph the six Tests/Quizzes Text Lectures Graphing calculator 2.1.PC.8 circular functions. Homework Worksheets Discussion SmartBoard Class Participation Investigations Links to “Flipped Classroom” Investigations Activities resources: (e.g. Khan Academy)

Write the value of an Tests/Quizzes Text Lectures Graphing calculator 2.2.PC.1 inverse trigonometric Homework Worksheets Discussion SmartBoard expression in radians Class Participation Investigations Links to “Flipped Classroom” Investigations Activities resources: (e.g. Khan Academy) Determine the angular Tests/Quizzes Text Lectures Graphing calculator 2.2.PC.4 and linear velocities of an Homework Worksheets Discussion SmartBoard object moving at a Class Participation Investigations Links to “Flipped Classroom” constant speed on a Investigations Activities resources: (e.g. Khan Academy) circular path Evaluate a trigonometric Tests/Quizzes Text Lectures Graphing calculator 2.2.PC.5 expression using radian Homework Worksheets Discussion SmartBoard measure. Class Participation Investigations Links to “Flipped Classroom” Investigations Activities resources: (e.g. Khan Academy)

Unit 2 Topic/Content Skills Assessment Materials Instructional Method Tech Infusion NJCCCS/CC Convert degree measure Tests/Quizzes Text Lectures Graphing calculator 2.2.PC.6 to radian measure. Homework Worksheets Discussion SmartBoard Class Participation Investigations Links to “Flipped Classroom” Investigations Activities resources: (e.g. Khan Academy)

6 Measure indirectly using Tests/Quizzes Text Lectures Graphing calculator 2.2.PC.7 trigonometric Homework Worksheets Discussion SmartBoard relationships. Class Participation Investigations Links to “Flipped Classroom” Investigations Activities resources: (e.g. Khan Academy)

Differentiated Instruction: Strategic Learner: Use visual models and manipulatives. Use graphing calculator and SmartBoard. Advanced Learner: Emphasize proof and foundations.

Ethical Decision Making/Character Education Review Honor Code and its implication for Midterm exams. Discuss/review policy on group work vs. individual.

21st Century Skills Working in teams, students study examples of Kolams from Southern Indiaor Sona from Angola and the Congoand explore the mathematics in these complex geometric art forms. They will explore Kolam designs based on Fibonacci numbers (http://vindhiya.com/Naranan/Fibonacci- Kolams/ ) and examine how an array of dots can give rise to a one-line sona (www.beloit.edu/computerscience/faculty/chavey/sona/). Using the concepts they have learned, students try to create their own Kolams or Sona.

Unit III: Trigonometric Identities Essential Question: How are the six trigonometric and circular functions related to each other?

Objectives: Students will be able to  Represent functions and relations numerically, graphically, and algebraically.  Use numerical, algebraic, and graphical representations of functions and relations in order to solve real-world problems.  Apply geometric properties and relationships to solve problems using tools and technology when appropriate. 7 Topic/Content Skills Assessment Materials/Resources Instructional Method Tech Infusion NJCCCS/CC Apply the fundamental Tests/Quizzes Text Lectures Graphing calculator 1.1.PC.17 trigonometric identities. Homework Worksheets Discussion SmartBoard Class Participation Investigations Links to “Flipped Investigations Activities Classroom” resources: (e.g. Khan Academy) Solve trigonometric Tests/Quizzes Text Lectures Graphing calculator 1.4.PC.5 equations. Homework Worksheets Discussion SmartBoard Class Participation Investigations Links to “Flipped Investigations Activities Classroom” resources: (e.g. Khan Academy) Interpret and solve Tests/Quizzes Text Lectures Graphing calculator 1.4.PC.10 problems involving Homework Worksheets Discussion SmartBoard trigonometric functions. Class Participation Investigations Links to “Flipped Investigations Activities Classroom” resources: (e.g. Khan Academy) Evaluate a trigonometric Tests/Quizzes Text Lectures Graphing calculator 2.2.PC.5 expression using radian Homework Worksheets Discussion SmartBoard measure. Class Participation Investigations Links to “Flipped Investigations Activities Classroom” resources: (e.g. Khan Academy) Measure indirectly using Tests/Quizzes Text Lectures Graphing calculator 2.2.PC.7 trigonometric Homework Worksheets Discussion SmartBoard relationships. Class Participation Investigations Links to “Flipped Investigations Activities Classroom” resources: (e.g. Khan Academy)

Differentiated Learning Activities Use the graphic calculator and smartboard to explore various graphs and manipulations of the graphs.

21st Century Skills Students explore knot theoryand its role in genetics (www.oglethorpe.edu/faculty/~j_nardo/knots/history.htm ). They learn how the DNA in every cell of our bodies is made up of long, coiled strands that get mathematically knotted in ways that lead to chemical changesin the DNA. Using string, rope, or even tangled audio headphone chords, students create several types of knots, such as those shown at www.mathedpage.org/pcmi/knot-theory.pdf . Then students examine each other’s knots and identify them by type: unknot, trefoil, figure eight,

8 five-star, one of the six’s, etc. Students photograph and label the knots and display them by creating a poster or posting them to a photo gallery on a class website.

Unit IV: Rational Functions Essential Question: How do rational functions model real-world problems and their solutions?

Objectives: Students will be able to  Represent functions and relations numerically, graphically, and algebraically.  Describe and apply properties of functions and relations.  Use numerical, algebraic, and graphical representations of functions and relations in order to solve real-world problems

Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS/CC Write a rational function Tests/Quizzes Text Lectures Graphing calculator 1.1.PC.2 or expression in an Homework Worksheets Discussion SmartBoard equivalent form, Class Participation Investigations Links to “Flipped including partial Investigations Activities Classroom” resources: fractions. (e.g. Khan Academy)

Graph rational functions Tests/Quizzes Text Lectures Graphing calculator 1.1.PC.11 and describe their Homework Worksheets Discussion SmartBoard properties. Class Participation Investigations Links to “Flipped Investigations Activities Classroom” resources: (e.g. Khan Academy)

Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS/CC Graph rational functions Tests/Quizzes Text Lectures Graphing calculator 1.1.PC.12 and describe their Homework Worksheets Discussion SmartBoard properties, including Class Participation Investigations Links to “Flipped limit theory as it applies Investigations Activities Classroom” resources: to determining their (e.g. Khan Academy) asymptotes & removable discontinuities. 9 Describe the properties Tests/Quizzes Text Lectures Graphing calculator 1.2.PC.1 of rational functions, Homework Worksheets Discussion SmartBoard including domain, range, Class Participation Investigations Links to “Flipped continuity, end behavior, Investigations Activities Classroom” resources: horizontal and vertical (e.g. Khan Academy) asymptotes. Describe oblique Tests/Quizzes Text Lectures Graphing calculator 1.2.PC.2 asymptotes of rational Homework Worksheets Discussion SmartBoard functions. Class Participation Investigations Links to “Flipped Investigations Activities Classroom” resources: (e.g. Khan Academy) Identify and distinguish Tests/Quizzes Text Lectures Graphing calculator 1.2.PC.6 between the graphs of Homework Worksheets Discussion SmartBoard linear, quadratic, power, Class Participation Investigations Links to “Flipped polynomial, rational, Investigations Activities Classroom” resources: exponential, logarithmic, (e.g. Khan Academy) trigonometric, and inverse trigonometric functions. Solve rational equations Tests/Quizzes Text Lectures Graphing calculator 1.4.PC.3 numerically, graphically, Homework Worksheets Discussion SmartBoard or algebraically. Class Participation Investigations Links to “Flipped Investigations Activities Classroom” resources: (e.g. Khan Academy) Solve rational Tests/Quizzes Text Lectures Graphing calculator 1.4.PC.4 inequalities using a Homework Worksheets Discussion SmartBoard numeric method. Class Participation Investigations Links to “Flipped Investigations Activities Classroom” resources: (e.g. Khan Academy) Differentiated Learning Activities Strategic Learner: Break problems into distinct steps, verbalizing the reasoning process. Encourage the use of the graphing calculator. Students should refer to Khan tutorials for additional support. Advanced Learner: Pose journal problems. Discuss ill-posed problems. Students may create their own tutorials to help peers and to demonstrate their own mastery.

21st Century Skills Students work in groups to investigate the history of a topic within the class’s current field of study, using websites such as the MacTutor History of Mathematics (http://www-history.mcs.st-and.ac.uk/ ). For example, if the class is studying algebra, each group explores the history of an 10 algebra topic, such as quadratic equations, group theory, or set theory. Each group creates a wiki entry based on its findings.

Unit V: Exponential and Logarithmic Functions Essential Question: How do exponential functions model real-world problems and their solutions?

Objectives: Students will be able to  Represent functions and relations numerically, graphically, and algebraically.  Describe and apply properties of functions and relations.  Perform a variety of operations and geometrical transformations on functions and relations.  Use numerical, algebraic, and graphical representations of functions and relations in order to solve real-world problems.  Estimate and compute using mental strategies, paper and pencil, and technology.

Topic/Content Skills Assessment Materials/Resources Instructional Method Tech Infusion NJCCCS/CC Write an exponential Tests/Quizzes Text Lectures Graphing calculator 1.1.PC.3 function or expression in Homework Worksheets Discussion SmartBoard an equivalent form using Class Participation Investigations Links to “Flipped laws of exponents. Investigations Activities Classroom” resources: (e.g. Khan Academy)

Topic/Content Skills Assessment Materials/Resources Instructional Method Tech Infusion NJCCCS Describe the properties of Tests/Quizzes Text Lectures Graphing calculator 1.2.PC.3 linear, quadratic, power, Homework Worksheets Discussion SmartBoard polynomial, rational, Class Participation Investigations Links to “Flipped exponential, logarithmic, Investigations Activities Classroom” resources: trigonometric, and inverse (e.g. Khan Academy) trigonometric functions. 11 Identify and distinguish Tests/Quizzes Text Lectures Graphing calculator 1.2.PC.6 between the graphs of Homework Worksheets Discussion SmartBoard linear, quadratic, power, Class Participation Investigations Links to “Flipped polynomial, rational, Investigations Activities Classroom” resources: exponential, logarithmic, (e.g. Khan Academy) trigonometric, and inverse trigonometric functions. Describe the effect of Tests/Quizzes Text Lectures Graphing calculator 1.3.PC.2 transformations on graphs Homework Worksheets Discussion SmartBoard of exponential functions, f Class Participation Investigations Links to “Flipped (x)= a(b)cx. Investigations Activities Classroom” resources: (e.g. Khan Academy) Describe the effect of Tests/Quizzes Text Lectures Graphing calculator 1.3.PC.3 transformations on graphs Homework Worksheets Discussion SmartBoard of logarithmic functions. Class Participation Investigations Links to “Flipped Investigations Activities Classroom” resources: (e.g. Khan Academy) Solve exponential Tests/Quizzes Text Lectures Graphing calculator 1.4.PC.1 equations, including base Homework Worksheets Discussion SmartBoard e, using various methods Class Participation Investigations Links to “Flipped including laws of Investigations Activities Classroom” resources: logarithms. (e.g. Khan Academy) Solve logarithmic Tests/Quizzes Text Lectures Graphing calculator 1.4.PC.2 equations, including base Homework Worksheets Discussion SmartBoard e, using laws of logarithms Class Participation Investigations Links to “Flipped and exponents. Investigations Activities Classroom” resources: (e.g. Khan Academy)

Topic/Content Skills Assessment Materials/Resources Instructional Method Tech Infusion NJCCCS Interpret and solve Tests/Quizzes Text Lectures Graphing calculator 1.4.PC.7 problems involving Homework Worksheets Discussion SmartBoard exponential functions. Class Participation Investigations Links to “Flipped Investigations Activities Classroom” resources: (e.g. Khan Academy)

12 Interpret and solve Tests/Quizzes Text Lectures Graphing calculator 1.4.PC.8 problems involving Homework Worksheets Discussion SmartBoard logarithmic functions. Class Participation Investigations Links to “Flipped Investigations Activities Classroom” resources: (e.g. Khan Academy) Evaluate a logarithm using Tests/Quizzes Text Lectures Graphing calculator 4.2.PC.5 the change of base rule. Homework Worksheets Discussion SmartBoard Class Participation Investigations Links to “Flipped Investigations Activities Classroom” resources: (e.g. Khan Academy)

Differentiated Learning Activities Collaborate with a science class and watch cell division.

Ethical Decision Making/Character Education Discuss the concepts of loans and interest in banking and investments.

21st Century Skills: Problem-Based Assignment Students review the algebraic formula for exponential growth and use the formula to solve a basic problem involving the spread of a viral disease: If 100 people are currently infected, and the number of people infected doubles every twelve hours, how many people will be infected after one week? The class discusses the exponential spread of infections and identifies other relevant examples of exponential growth, such as: bacteria growth; human population growth; the growth of atmospheric carbon dioxide; and in finance, the compounding of interest. Then students work in groups: each group poses a question involving exponential growth and gives it to another group to answer. Each group shares its results with the class.

Unit VI: Vectors, Parametrics, and Polars Essential Question: Why are functions and relations represented by parametric equations?

Objectives: Students will be able to  Represent functions and relations numerically, graphically, and algebraically.  Use numerical, algebraic, and graphical representations of functions and relations in order to solve real-world problems. 13  Analyze two- and three-dimensional figures using tools and technology when appropriate.  Apply geometric properties and relationships to solve problems using tools and technology when appropriate.  Describe and represent numbers and their relationships.  Estimate and compute using mental strategies, paper and pencil, and technology

Topic/Content Skills Assessment Materials/Resources Instructional Method Tech Infusion NJCCCS Write equivalent forms for Tests/Quizzes Text Lectures Graphing calculator 1.1.PC.4 rectangular and polar Homework Worksheets Discussion SmartBoard equations. Class Participation Investigations Links to “Flipped Investigations Activities Classroom” resources: (e.g. Khan Academy) Write equivalent equations Tests/Quizzes Text Lectures Graphing calculator 1.1.PC.5 for functions and relations Homework Worksheets Discussion SmartBoard in parametric and Class Participation Investigations Links to “Flipped rectangular form. Investigations Activities Classroom” resources: (e.g. Khan Academy) Write a vector equation of Tests/Quizzes Text Lectures Graphing calculator 1.1.PC.6 a line in 2-space. Homework Worksheets Discussion SmartBoard Class Participation Investigations Links to “Flipped Investigations Activities Classroom” resources: (e.g. Khan Academy) Write an equation of a line Tests/Quizzes Text Lectures Graphing calculator 1.1.PC.7 or plane in 3-space. Homework Worksheets Discussion SmartBoard Class Participation Activities Links to “Flipped Investigations Classroom” resources: (e.g. Khan Academy) Graph polar equations and Tests/Quizzes Text Lectures Graphing calculator 1.1.PC.13 describe their properties. Homework Worksheets Discussion SmartBoard Class Participation Activities Investigations

Topic/Content Skills Assessment Materials/Resources Instructional Method Tech Infusion NJCCCS Graph a point, line, or Tests/Quizzes Text Lectures Graphing calculator 1.1.PC.14 plane in 3-space. Homework Worksheets Discussion SmartBoard Class Participation Investigations Links to “Flipped Investigations Activities Classroom” resources: (e.g. Khan Academy) 14 Solve systems of Tests/Quizzes Text Lectures Graphing calculator 1.4.PC.6 equations in polar form. Homework Worksheets Discussion SmartBoard Class Participation Investigations Links to “Flipped Investigations Activities Classroom” resources: (e.g. Khan Academy) Interpret and solve Tests/Quizzes Text Lectures Graphing calculator 1.4.PC.11 problems involving Homework Worksheets Discussion SmartBoard parametric functions and Class Participation Investigations Links to “Flipped relations. Investigations Activities Classroom” resources: (e.g. Khan Academy) Write equivalent Tests/Quizzes Text Lectures Graphing calculator 2.1.PC.1 rectangular and polar Homework Worksheets Discussion SmartBoard forms of points on the Class Participation Activities Links to “Flipped coordinate plane. Investigations Classroom” resources: (e.g. Khan Academy) Represent a vector in 2- Tests/Quizzes Text Lectures Graphing calculator 2.1.PC.2 space by its magnitude Homework Worksheets Discussion SmartBoard and direction, its initial Class Participation Activities Links to “Flipped and terminal point, and its Investigations Classroom” resources: component form. (e.g. Khan Academy) Represent a vector in 3- Tests/Quizzes Text Lectures Graphing calculator 2.1.PC.3 space by its magnitude Homework Worksheets Discussion SmartBoard and direction, its initial Class Participation Activities Links to “Flipped and terminal point, and its Investigations Classroom” resources: component form. (e.g. Khan Academy) Determine multiple polar Tests/Quizzes Text Lectures Graphing calculator 2.1.PC.6 form representations of a Homework Worksheets Discussion SmartBoard point. Class Participation Activities Investigations Topic/Content Skills Assessment Materials/Resources Instructional Method Tech Infusion NJCCCS Identify the pole and the Tests/Quizzes Text Lectures Graphing calculator 2.1.PC.7 polar axis, and plot points Homework Worksheets Discussion SmartBoard given in polar form. Class Participation Investigations Links to “Flipped Investigations Activities Classroom” resources: (e.g. Khan Academy)

15 Determine the distance Tests/Quizzes Text Lectures Graphing calculator 2.2.PC.2 from a point to a line in 2- Homework Worksheets Discussion SmartBoard space. Class Participation Investigations Links to “Flipped Investigations Activities Classroom” resources: (e.g. Khan Academy) Write equivalent Tests/Quizzes Text Lectures Graphing calculator 4.1.PC.1 rectangular and polar Homework Worksheets Discussion SmartBoard forms for complex Class Participation Activities Links to “Flipped numbers. Investigations Represent complex Tests/Quizzes Text Lectures Graphing calculator 4.1.PC.2 numbers in polar form Homework Worksheets Discussion SmartBoard numerically and Class Participation Activities Links to “Flipped graphically. Investigations Classroom” resources: (e.g. Khan Academy) Determine the product or Tests/Quizzes Text Lectures Graphing calculator 4.2.PC.1 quotient of two complex Homework Worksheets Discussion SmartBoard numbers in polar form. Class Participation Activities Determine a power or the Tests/Quizzes Text Lectures Graphing calculator 4.2.PC.2 roots of a complex number Homework Worksheets Discussion SmartBoard using DeMoivre’s Class Participation Activities theorem. Investigations Determine the sum, Tests/Quizzes Text Lectures Graphing calculator 4.2.PC.3 difference, scalar product, Homework Worksheets Discussion SmartBoard and dot product of vectors Class Participation Activities in 2-space. Investigations Determine the sum, Tests/Quizzes Text Lectures Graphing calculator 4.2.PC.4 difference, scalar product, Homework Worksheets Discussion SmartBoard dot product & cross Class Participation Activities Links to “Flipped product of vectors in 3- Investigations Classroom” resources: space. (e.g. Khan Academy)

Differentiated Learning Activities Strategic Learner: Break problems into distinct steps, verbalizing the reasoning process. Encourage the use of the graphing calculator. Students should refer to Khan tutorials for additional support. Advanced Learner: Pose journal problems. Discuss ill-posed problems. Students may create their own tutorials to help peers and to demonstrate their own mastery.

21st Century Skills: Problem-Based Assignment 16 Students explore the mathematics of the popular puzzle video game Angry Birds (http://forum.davidson.edu/mathmovement/2011/11/13/algebra- of-angry- birds/ ).Working in teams, students build a hands- on version of the game. They create targets by stacking blocks or other lightweight objects and they construct a toy catapult or slingshot to aim at the target (or toss a beanbag). After experimenting by trying to knock down the target from various starting points, students come up with different equations for parabolas to see which will work to knock down the structure from a given distance.

Unit VII: Discrete Math Essential Question: How do sequences and series model real-world problems and their solutions?

Objectives: Students will be able to • Represent functions and relations numerically, graphically, and algebraically • Estimate compute using mental strategies, paper and pencil, and technology

Topic/Content Skills Assessment Materials/Resources Instructional Method Tech Infusion NJCCCS Represent a series using Tests/Quizzes Text Lectures Graphing calculator 1.1.PC.10 summation notation. Homework Worksheets Discussion SmartBoard Class Participation Investigations Links to “Flipped Investigations Activities Classroom” resources: (e.g. Khan Academy) Determine the sum, if it Tests/Quizzes Text Lectures Graphing calculator 1.1.PC.16 exists, of an infinite Homework Worksheets Discussion SmartBoard geometric series. Class Participation Investigations Links to “Flipped Investigations Activities Classroom” resources: (e.g. Khan Academy)

Topic/Content Skills Assessment Materials/Resources Instructional Method Tech Infusion NJCCCS Expand and evaluate a Tests/Quizzes Text Lectures Graphing calculator 1.1.PC.18 series written in Homework Worksheets Discussion SmartBoard summation notation. Class Participation Investigations Links to “Flipped Investigations Activities Classroom” resources: (e.g. Khan Academy)

17 Expand powers of Tests/Quizzes Text Lectures Graphing calculator 4.2.PC.6 binomials applying the Homework Worksheets Discussion SmartBoard binomial theorem, Class Participation Investigations Links to “Flipped factorials & Investigations Activities Classroom” resources: combinatorics. (e.g. Khan Academy)

Differentiated Learning Activities Discuss the role of sequences in chaos theory, Julia Sets, etc. Cross-curricular study physics.

21st Century Skills: Problem-Based Assignment Students form teams to engage in a modeling project similar to problems presented in the Mathematical Contest in Modeling run each year by COMAP. For example, the following problem is adapted from the 2009 competition:

Many cities and communities have traffic circles. Some traffic circles are large and have many lanes, such as at the Arc de Triomphe in Paris, and the Victory Monument in Bangkok. Some have only one or two lanes in the circle. Some traffic circles position a stop sign or a yield sign on every incoming road that gives priority to traffic already in the circle. Others post yield signs in the circle at each incoming road to give priority to incoming traffic. Still others have a traffic light where each incoming road meets the circle (with no right turn allowed on a red light). Other designs may also be possible.

Students form teams and each team develops a model for creating a traffic circle in their own community or in a nearby city (or modifying an existing traffic circle). Here are the directions:  First, team members discuss how many lanes should be in the circle and the rate of traffic flow. If feasible, they may collect actual traffic data. Team members outline alternatives for the placement of stop signs or yield signs and other design factors. Then they work together to determine how to best control traffic flow in, around, and out of the circle.  Team members determine the best alternatives and explain what factors influenced their decision, such as the effect on traffic flow and safety.  Team members refine their model. For example, if traffic lights are recommended, students develop a method for determining how many seconds each light should remain green, which may vary according to the time of day and other factors.