presentation: module focus algebra ii module 2

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NYS COMMON CORE MATHE MATICS CURRI CULUM A Story of Functions

A Story of FunctionsA Close Look at Algebra II Module 2


N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Functions

Opening ExerciseImagine a point moving along a square with “radius” of 1 centered about the origin (meaning that 1 is half of the side length of the square).At each angle, the object has a height that we will call square sine or squine(x).

Draw the graph of y = squine(x).



Participant Poll• Classroom teacher• Math trainer• Principal or school leader• District representative / leader• Other



Session Objectives• Experience and model the instructional approaches to

teaching the content of Algebra II Module 2 lessons. • Articulate how the lessons promote mastery of the focus

standards and how the module addresses the major work of the grade.

• Make connections from the content of previous modules and grade levels to the content of this module.



Agenda• Overview of Module 2• Topic A • Mid-Module Assessment• Topic B• End of Module Assessment



Flow of Module 2• 17 lessons • 7 exploration• 6 socratic• 4 problem set

• 21 instructional days• Topic A: The Story of Trigonometry and Its Context• Topic B: Understanding Trigonometric Functions and Putting

Them to Use



Mathematical Themes of Module 2• Emphasis on exploratory lessons rather than problem set lessons

• Developing the sine and cosine functions through exploration of the horizontal and vertical displacement of

a car on a rotating Ferris wheel

• Defining sine and cosine as functions of real numbers

• Graphing the the sine and cosine functions and transformations of these functions

• Discovering and proving trigonometric identities



What are students coming in with?• Experience with graphing transformations of functions on the

coordinate plane (G9)• Experience modeling data using appropriate functions (G9 and

G11)• Experience with trigonometric ratios and circles (G10)



Topic A: The Story of Trigonometry and Its Contexts • The six trigonometric functions are defined as functions of the

amount of rotation of a point on the unit circle from the positive x-axis .

• Students are given the historical context of how trigonometry was developed.

• The values of the sine and cosine functions for multiples of , , and are used to generate the graphs of the sine and cosine functions.

• The graphs of the sine and cosine function are used to discover basic trigonometric identities.



Lessons 1 and 2: Ferris Wheels – Tracking the Height and Co-Height of a Passenger Car • Students model the circular motion of a Ferris wheel using a

paper plate. • Sine and cosine are not mentioned in these lessons. • The functions are described as the “height and co-height

functions.”• The lessons employ a familiar scenario to introduce the ideas

of circular motion and periodic functions.



Lessons 1 and 2: Ferris Wheels – Tracking the Height and Co-Height of a Passenger Car

PERIODIC FUNCTION. A function f whose domain is a subset of the real numbers is said to be periodic with period P if the domain of f contains x+P whenever it contains x, and if f(x+P)=f(x) for all real numbers x in its domain. If a least positive number P exists that satisfies this equation, it is called the fundamental period, or if the context is clear, just the period of the function.



Lessons 3: The Motion of the Moon, Sun, and Stars—Motivating Mathematics



Lessons 4: From Circle-ometry to Trigonometry• Work example 1 and 2.• Work problem set 2 – 5.• Lots of information to process in this lesson.• Consider using the scaffold suggestion of creating a graphic

organizer.



Lessons 4: From Circle-ometry to Trigonometry



A “Handy” Trick



Lessons 5: Extending the Domain of Sine and Cosine to All Real Numbers• Students explore clockwise rotations and rotations greater

than 360 degrees• Students also explore rotations in which the terminal ray lands

on the axis or axis.• Work through the discussion case 1 and 2.



Lesson 6: Why Call it Tangent?• Work the opening exercise.• You may need to review some terminology from geometry.



Lesson 6: Why Call it Tangent?• Students interpret the tangent function several ways:



Lesson 7: Secant and the Co-Functions• What is the geometric meaning of the secant function?• Why is cosecant the reciprocal of the sine function rather than the cosine

function?• How do we make sense of the domain and range of the various

trigonometric functions?• Do we really need these three trigonometric functions?



Sprint

Evaluating trigonometric functions



Lesson 8: Graphing the Sine and Cosine Functions • Students construct the graphs of f and g first by using the vertical and

horizontal components of points on the unit circle and then by using the known values of sine and cosine.

• The emphasis should be on the key features of each graph.• Relative maxima and minima• Intervals of increasing and decreasing• Intercepts• End behavior• Symmetry• Amplitude• Period



Lesson 9: Awkward! Who Chose the Number 360 Anyway?• Why do we need radian measurement when degrees have worked fine up

to this point?• Arc length calculations are easier for radian measurements.• When graphing sine and cosine functions, the vertical and horizontal axes must be

scaled completely differently.



Lesson 9: Awkward! Who Chose the Number 360 Anyway?• The main reason that we see the transition from degrees to radians is that

all trigonometry done in Calculus is based on radian measurement.

• Find .

• In Calculus, the fact that Is used to derive the fact that the derivative of sin(x) is cos(x) when we measure x in radians.

xx

x

)sin(lim0



Lesson 9: Awkward! Who Chose the Number 360 Anyway?



Lesson 10: Basic Trigonometric Identities from Graphs• Students use the graphs of the sine and cosine functions to note properties

that lead to trigonometric identities.• Periodicity• Symmetry (even/odd functions)• Horizontal translations

• Note that an identity is a statement that two functions are equal on a common domain. • The statement “sin(x+2π)=sin(x)” itself is not an identity.• The correct statement is “sin(x+2π)=sin(x) for all real numbers x.”



Lesson 10: Basic Trigonometric Identities from Graphs• Key takeaways from this lesson:• The basic trigonometric identities derived from the periodic nature and symmetry

of the graphs of each function. • For all real numbers x,

• The graphs of sine and cosine can be made to coincide with one another by using horizontal shifts.• For all real numbers x,



Key Points – Topic A• There is a rich historical context that motivates the study of

trigonometry. • Students are given opportunities to explore the meaning of the

trigonometric functions and their properties in order to develop a deeper understanding.

• Students make connections between the values of sine and cosine for particular amounts of rotation with the graphs of the functions and then to trigonometric identities.



Mid-Module Assessment

Work with a partner on this assessment



Scoring the Assessment



Topic B: Understanding Trigonometric Functions and Putting Them to Use • Students use trigonometric functions to model periodic behavior.• The concept of polynomial identities covered in module 1 is extended to

trigonometric identities.• Continue to emphasize the proper statement of a trigonometric identity as

the pairing of a statement that two functions are equivalent on a given domain and an identification of that domain.



Lesson 11: Transforming the Graph of the Sine Function• Students explore the family of functions given by the generalized sine

function .• Students are placed in 4 teams, and each team is responsible for

determining the effect of changing one of the four parameters ( and ) on the graph of the sine function.



Lesson 11: Transforming the Graph of the Sine Function• For a sine function written in the form , for real numbers ,,, and • is called the amplitude of the function.• is the period of the function.• is the frequency of the function (the frequency is the reciprocal of the period).• is called the phase shift. • the graph of is called the midline.



Lesson 12: Ferris Wheels—Using Trigonometric Functions to Model Cyclical Behavior• Work through the exploratory challenge (exercises 1 – 5).• Consider the height function from exercise 4: • How would this formula change for a Ferris wheel with a different diameter?• How would this formula change for a Ferris wheel at a different height off the

ground?• How would this formula change for a Ferris wheel that had a different rate of

revolution?• How would this formula change if we modeled the height of a passenger car above

the ground from a different starting position on the wheel?



Lesson 13: Tides, Sound Waves, and Stock Markets



Lesson 14: Graphing the Tangent Function• Students work in groups where each group is responsible for creating one

branch of the graph of the tangent function.• These branches are then combined into a single graph.• Students then work on trigonometric identities involving tangent.• Continue to be precise when stating an identity.• for , for all integers

• Note that is used when working on a circle and is used when working on a coordinate plane.



Lesson 14: Graphing the Tangent Function• At this point, it would be valuable to compare identities developed

throughout this module.



Lesson 15: What is a Trigonometric Identity?• The lesson opens by developing the

Pythagorean Identity.• Since and , it follows that .• Students then derive a second identity

from the Pythagorean Theorem.• for all real numbers such that , for all integers



Lesson 16: Proving Trigonometric Identities• Is the following statement a trigonometric identity for all • What is wrong with this mathematical proof?First, [1] for any real number. Using the multiplication property of equality, square both sides, which gives

[2] for any real number.Using the subtraction property of equality, subtract from each side, which

gives [3] for any real number.Statement [3] is the Pythagorean identity. So, replace by to get [4] , which is

definitely true.Therefore, the original statement must be true.



Lesson 17: Trigonometric Identity Proofs• Wishful thinking: By any chance, does ?• Students first find a formula by exploring patterns (MP 8).• They then prove their conjecture for two positive numbers whose sum is

less than .• Work Examples 1 and 2.



Key Points – Topic B• Trigonometric functions are useful for modeling periodic data.• A trigonometric identity is a statement that two functions are

equivalent on a given domain and an identification of that domain.

• Trigonometric identities can be proven graphically, numerically, and algebraically.



Key Points – Module 2 Lessons



End-of-Module Assessment

Work with a partner on this assessment



Scoring the Assessment



Key Points – End-of-Module Assessment• End of Module assessment are designed to assess all standards

of the module (at least at the cluster level) with an emphasis on assessing thoroughly those presented in the second half of the module.

• Recall, as much as possible, assessment items are designed to asses the standards while emulating PARCC Type 2 and Type 3 tasks.

• Recall, rubrics are designed to inform each district / school / teacher as they make decisions about the use of assessments in the assignment of grades.



Biggest Takeaway

What are your biggest takeaways from the study of Module 2?

How can you support successful implementation of these materials at your schools given your role as a teacher, trainer, school or district leader, administrator or other representative?

presentation: module focus algebra ii module 2

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