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 Represent a piece-wise function numerically, algebraically, and graphically. Assessment Tests/Quizzes Homework Class Participation Class Investigations Resources Describe the properties of linear, quadratic, power, polynomial, rational, exponential, logarithmic, trigonometric, and inverse trigonometric functions Describe the properties of a piece-wise function. Tests/Quizzes Homework Class Participation Class Investigations Text Worksheets Tests/Quizzes Homework Class Participation Class Investigations Text Worksheets Text Worksheets 2 Instructional Method Lectures Discussion Investigations Activities Tech Infusion Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard NJCCCS/CC 1.1.PC.8 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.2.PC.3 Links to “Flipped Classroom” resources: (e.g. Khan Academy) Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.2.PC.4 Topic/Content Skills Assessment Resources Identify and distinguish between the graphs of linear, quadratic, power, polynomial, rational, exponential, logarithmic, trigonometric, and inverse trigonometric functions. Tests/Quizzes Homework Class Participation Class Investigations Text Worksheets Describe the effect of single or multiple transformations on the graph of f(x), including a f(x), f⎛x−h⎞, f(x)+k, f(ax), f x, and f⎛x⎞, f(|x|), |f(x)| Describe the effect of transformations on a function with a restricted domain. Tests/Quizzes Homework Class Participation Class Investigations Text Worksheets Tests/Quizzes Homework Class Participation Class Investigations Text Worksheets Interpret and solve problems involving piece-wise functions. Tests/Quizzes Homework Class Participation Class Investigations Text Worksheets Make predictions using trigonometric or power mathematical models given a set of data. Tests/Quizzes Homework Class Participation Class Investigations Text Worksheets Instructional Method 3 Tech Infusion Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard NJCCCS/CC 1.2.PC.6 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.3.PC.1 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.3.PC.5 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.4.PC.9 Links to “Flipped Classroom” resources: (e.g. Khan Academy) Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.4.PC.12 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? 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 Determine the period, amplitude, phase shift, and/or vertical shift of a trigonometric function represented graphically or algebraically. Assessment Materials Tests/Quizzes Text Homework Worksheets Class Participation Class Investigations Instructional Method Lectures Discussion Investigations Activities 4 Tech Infusion Graphing calculator SmartBoard Links to “Flipped Classroom” resources: (e.g. Khan Academy) NJCCCS/CC 1.1.PC.15 Unit 2 Topic/Content Skills Describe the properties of linear, quadratic, power, polynomial, rational, exponential, logarithmic, trigonometric, and inverse trigonometric functions. Describe the inverse relationship between trigonometric and inverse trigonometric functions, numerically, algebraically, and graphically. Identify and distinguish between the graphs of linear, quadratic, power, polynomial, rational, exponential, logarithmic, trigonometric, and inverse trigonometric functions Describe the effect of transformations on the graphs of trigonometric functions. Interpret and solve problems involving trigonometric functions. Assessment Tests/Quizzes Homework Class Participation Investigations Materials Text Worksheets Tests/Quizzes Homework Class Participation Investigations Text Worksheets Tests/Quizzes Homework Class Participation Investigations Text Worksheets Tests/Quizzes Homework Class Participation Investigations Text Worksheets Tests/Quizzes Homework Class Participation Investigations Text Worksheets 5 Instructional Method Lectures Discussion Investigations Activities Tech Infusion Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard NJCCCS/CC 1.2.PC.3 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.2.PC.5 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.2.PC.6 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.3.PC.4 Links to “Flipped Classroom” resources: (e.g. Khan Academy) Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.4.PC.10 Unit 2 Topic/Content Skills Make predictions using trigonometric or power mathematical models given a set of data.. Assessment Tests/Quizzes Homework Class Participation Investigations Materials Text Worksheets Tests/Quizzes Homework Class Participation Investigations Text Worksheets Tests/Quizzes Homework Class Participation Investigations Text Worksheets Tests/Quizzes Homework Class Participation Investigations Text Worksheets Write the value of an inverse trigonometric expression in radians Tests/Quizzes Homework Class Participation Investigations Text Worksheets Determine the angular and linear velocities of an object moving at a constant speed on a circular path Evaluate a trigonometric expression using radian measure. Tests/Quizzes Homework Class Participation Investigations Text Worksheets Tests/Quizzes Homework Class Participation Investigations Text Worksheets Describe and apply the relationship between the trigonometry of the right triangle and the unit circle. Describe and apply the relationship between the radian measure of a central angle of a circle and its intercepted arc. Define and graph the six circular functions. 6 Instructional Method Lectures Discussion Investigations Activities Tech Infusion Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard NJCCCS/CC 1.4.PC.12 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 2.1.PC.4 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 2.1.PC.5 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 2.1.PC.8 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 2.2.PC.1 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 2.2.PC.4 Links to “Flipped Classroom” resources: (e.g. Khan Academy) Links to “Flipped Classroom” resources: (e.g. Khan Academy) 2.2.PC.5 Unit 2 Topic/Content Skills Convert degree measure to radian measure. Measure indirectly using trigonometric relationships. Assessment Tests/Quizzes Homework Class Participation Investigations Materials Text Worksheets Tests/Quizzes Homework Class Participation Investigations Text Worksheets Instructional Method Lectures Discussion Investigations Activities Tech Infusion Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard NJCCCS/CC 2.2.PC.6 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 2.2.PC.7 Links to “Flipped Classroom” 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 India or Sona from Angola and the Congo and explore the mathematics in these complex geometric art forms. They will explore Kolam designs based on Fibonacci numbers (http://vindhiya.com/Naranan/FibonacciKolams/ ) 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. 7 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. Topic/Content Skills Apply the fundamental trigonometric identities. Assessment Tests/Quizzes Homework Class Participation Investigations Materials/Resources Text Worksheets Tests/Quizzes Homework Class Participation Investigations Text Worksheets Interpret and solve problems involving trigonometric functions. Tests/Quizzes Homework Class Participation Investigations Text Worksheets Evaluate a trigonometric expression using radian measure. Tests/Quizzes Homework Class Participation Investigations Text Worksheets Measure indirectly using trigonometric relationships. Tests/Quizzes Homework Class Participation Investigations Text Worksheets Solve trigonometric equations. 8 Instructional Method Lectures Discussion Investigations Activities Tech Infusion Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard NJCCCS/CC 1.1.PC.17 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.4.PC.5 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.4.PC.10 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 2.2.PC.5 Links to “Flipped Classroom” resources: (e.g. Khan Academy) Links to “Flipped Classroom” resources: (e.g. Khan Academy) 2.2.PC.7 Differentiated Learning Activities Use the graphic calculator and smartboard to explore various graphs and manipulations of the graphs. 21st Century Skills Students explore knot theory and 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 changes in 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, 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 Write a rational function or expression in an equivalent form, including partial fractions. Assessment Tests/Quizzes Homework Class Participation Investigations Resources Text Worksheets Graph rational functions and describe their properties. Tests/Quizzes Homework Class Participation Investigations Text Worksheets 9 Instructional Method Lectures Discussion Investigations Activities Tech Infusion Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard NJCCCS/CC 1.1.PC.2 Links to “Flipped Classroom” resources: (e.g. Khan Academy) Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.1.PC.11 Topic/Content Skills Graph rational functions and describe their properties, including limit theory as it applies to determining their asymptotes & removable discontinuities. Describe the properties of rational functions, including domain, range, continuity, end behavior, horizontal and vertical asymptotes. Describe oblique asymptotes of rational functions. Assessment Tests/Quizzes Homework Class Participation Investigations Resources Text Worksheets Tests/Quizzes Homework Class Participation Investigations Text Worksheets Tests/Quizzes Homework Class Participation Investigations Text Worksheets Identify and distinguish between the graphs of linear, quadratic, power, polynomial, rational, exponential, logarithmic, trigonometric, and inverse trigonometric functions. Solve rational equations numerically, graphically, or algebraically. Tests/Quizzes Homework Class Participation Investigations Text Worksheets Tests/Quizzes Homework Class Participation Investigations Text Worksheets Solve rational inequalities using a numeric method. Tests/Quizzes Homework Class Participation Investigations Text Worksheets 10 Instructional Method Lectures Discussion Investigations Activities Tech Infusion Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard NJCCCS/CC 1.1.PC.12 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.2.PC.1 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.2.PC.2 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.2.PC.6 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.4.PC.3 Links to “Flipped Classroom” resources: (e.g. Khan Academy) Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.4.PC.4 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 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 Write an exponential function or expression in an equivalent form using laws of exponents. Assessment Tests/Quizzes Homework Class Participation Investigations Materials/Resources Text Worksheets 11 Instructional Method Lectures Discussion Investigations Activities Tech Infusion Graphing calculator SmartBoard Links to “Flipped Classroom” resources: (e.g. Khan Academy) NJCCCS/CC 1.1.PC.3 Topic/Content Skills Describe the properties of linear, quadratic, power, polynomial, rational, exponential, logarithmic, trigonometric, and inverse trigonometric functions. Identify and distinguish between the graphs of linear, quadratic, power, polynomial, rational, exponential, logarithmic, trigonometric, and inverse trigonometric functions. Describe the effect of transformations on graphs of exponential functions, f (x)= a(b)cx. Assessment Tests/Quizzes Homework Class Participation Investigations Materials/Resources Text Worksheets Tests/Quizzes Homework Class Participation Investigations Text Worksheets Tests/Quizzes Homework Class Participation Investigations Text Worksheets Describe the effect of transformations on graphs of logarithmic functions. Tests/Quizzes Homework Class Participation Investigations Text Worksheets Solve exponential equations, including base e, using various methods including laws of logarithms. Solve logarithmic equations, including base e, using laws of logarithms and exponents. Tests/Quizzes Homework Class Participation Investigations Text Worksheets Tests/Quizzes Homework Class Participation Investigations Text Worksheets 12 Instructional Method Lectures Discussion Investigations Activities Tech Infusion Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard NJCCCS 1.2.PC.3 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.2.PC.6 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.3.PC.2 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.3.PC.3 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.4.PC.1 Links to “Flipped Classroom” resources: (e.g. Khan Academy) Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.4.PC.2 Topic/Content Skills Interpret and solve problems involving exponential functions. Assessment Tests/Quizzes Homework Class Participation Investigations Materials/Resources Text Worksheets Interpret and solve problems involving logarithmic functions. Tests/Quizzes Homework Class Participation Investigations Text Worksheets Evaluate a logarithm using Tests/Quizzes the change of base rule. Homework Class Participation Investigations Text Worksheets Instructional Method Lectures Discussion Investigations Activities Tech Infusion Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard NJCCCS 1.4.PC.7 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.4.PC.8 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 4.2.PC.5 Links to “Flipped 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. 13 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. 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 Write equivalent forms for rectangular and polar equations. Assessment Tests/Quizzes Homework Class Participation Investigations Materials/Resources Text Worksheets Write equivalent equations for functions and relations in parametric and rectangular form. Tests/Quizzes Homework Class Participation Investigations Text Worksheets Write a vector equation of a line in 2-space. Tests/Quizzes Homework Class Participation Investigations Text Worksheets Tests/Quizzes Homework Class Participation Investigations Text Worksheets Tests/Quizzes Homework Class Participation Investigations Text Worksheets Write an equation of a line or plane in 3-space. Graph polar equations and describe their properties. 14 Instructional Method Lectures Discussion Investigations Activities Tech Infusion Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Activities Graphing calculator SmartBoard Lectures Discussion Activities Graphing calculator SmartBoard NJCCCS 1.1.PC.4 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.1.PC.5 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.1.PC.6 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.1.PC.7 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.1.PC.13 Topic/Content Skills Graph a point, line, or plane in 3-space. Assessment Tests/Quizzes Homework Class Participation Investigations Materials/Resources Text Worksheets Tests/Quizzes Homework Class Participation Investigations Text Worksheets Interpret and solve problems involving parametric functions and relations. Tests/Quizzes Homework Class Participation Investigations Text Worksheets Write equivalent rectangular and polar forms of points on the coordinate plane. Tests/Quizzes Homework Class Participation Investigations Text Worksheets Represent a vector in 2space by its magnitude and direction, its initial and terminal point, and its component form. Represent a vector in 3space by its magnitude and direction, its initial and terminal point, and its component form. Determine multiple polar form representations of a point. Tests/Quizzes Homework Class Participation Investigations Text Worksheets Tests/Quizzes Homework Class Participation Investigations Text Worksheets Tests/Quizzes Homework Class Participation Investigations Text Worksheets Solve systems of equations in polar form. 15 Instructional Method Lectures Discussion Investigations Activities Tech Infusion Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Activities Graphing calculator SmartBoard Lectures Discussion Activities Graphing calculator SmartBoard Lectures Discussion Activities Graphing calculator SmartBoard Lectures Discussion Activities Graphing calculator SmartBoard NJCCCS 1.1.PC.14 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.4.PC.6 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.4.PC.11 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 2.1.PC.1 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 2.1.PC.2 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 2.1.PC.3 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 2.1.PC.6 Topic/Content Skills Identify the pole and the polar axis, and plot points given in polar form. Assessment Tests/Quizzes Homework Class Participation Investigations Materials/Resources Text Worksheets Instructional Method Lectures Discussion Investigations Activities Tech Infusion Graphing calculator SmartBoard Determine the distance from a point to a line in 2space. Tests/Quizzes Homework Class Participation Investigations Text Worksheets Write equivalent rectangular and polar forms for complex numbers. Represent complex numbers in polar form numerically and graphically. Tests/Quizzes Homework Class Participation Investigations Tests/Quizzes Homework Class Participation Investigations Text Worksheets Lectures Discussion Investigations Activities Graphing calculator SmartBoard Lectures Discussion Activities Graphing calculator SmartBoard Lectures Discussion Activities Graphing calculator SmartBoard Determine the product or quotient of two complex numbers in polar form. Determine a power or the roots of a complex number using DeMoivre’s theorem. Determine the sum, difference, scalar product, and dot product of vectors in 2-space. Determine the sum, difference, scalar product, dot product & cross product of vectors in 3space. Tests/Quizzes Homework Class Participation Tests/Quizzes Homework Class Participation Investigations Tests/Quizzes Homework Class Participation Investigations Tests/Quizzes Homework Class Participation Investigations Text Worksheets Lectures Discussion Activities Lectures Discussion Activities Graphing calculator SmartBoard 4.2.PC.1 Graphing calculator SmartBoard 4.2.PC.2 Text Worksheets Lectures Discussion Activities Graphing calculator SmartBoard 4.2.PC.3 Text Worksheets Lectures Discussion Activities Graphing calculator SmartBoard 4.2.PC.4 Text Worksheets Text Worksheets 16 NJCCCS 2.1.PC.7 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 2.2.PC.2 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 4.1.PC.1 Links to “Flipped 4.1.PC.2 Links to “Flipped Classroom” resources: (e.g. Khan Academy) Links to “Flipped 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: Problem-Based Assignment 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 Represent a series using summation notation. Determine the sum, if it exists, of an infinite geometric series. Assessment Tests/Quizzes Homework Class Participation Investigations Materials/Resources Text Worksheets Tests/Quizzes Homework Class Participation Investigations Text Worksheets 17 Instructional Method Lectures Discussion Investigations Activities Tech Infusion Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard NJCCCS 1.1.PC.10 Links to “Flipped Classroom” resources: (e.g. Khan Academy) Links to “Flipped Classroom” resources: (e.g. Khan Academy) 1.1.PC.16 Topic/Content Skills Expand and evaluate a series written in summation notation. Assessment Tests/Quizzes Homework Class Participation Investigations Materials/Resources Text Worksheets Expand powers of binomials applying the binomial theorem, factorials & combinatorics. Tests/Quizzes Homework Class Participation Investigations Text Worksheets Instructional Method Lectures Discussion Investigations Activities Tech Infusion Graphing calculator SmartBoard Lectures Discussion Investigations Activities Graphing calculator SmartBoard NJCCCS 1.1.PC.18 Links to “Flipped Classroom” resources: (e.g. Khan Academy) 4.2.PC.6 Links to “Flipped Classroom” resources: (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. 18