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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
A Story of Functions
A Close Look at Algebra II Module 2
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Opening Exercise
Imagine a point moving along a square with โ€œradiusโ€ of 1 centered
about the origin (meaning that 1 is half of the side length of the
square).
At each angle, ๐‘ฅ, the object has a height that we will call square sine or
squine(x).
Draw the graph of y = squine(x).
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Participant Poll
โ€ข
โ€ข
โ€ข
โ€ข
โ€ข
Classroom teacher
Math trainer
Principal or school leader
District representative / leader
Other
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Session Objectives
โ€ข Experience and model the instructional approaches to
teaching the content of Algebra II Module 2 lessons.
โ€ข Articulate how the lessons promote mastery of the focus
standards and how the module addresses the major work of
the grade.
โ€ข Make connections from the content of previous modules and
grade levels to the content of this module.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
โ€ข
โ€ข
โ€ข
โ€ข
โ€ข
Overview of Module 2
Topic A
Mid-Module Assessment
Topic B
End of Module Assessment
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Flow of Module 2
โ€ข 17 lessons
โ€ข 7 exploration
โ€ข 6 socratic
โ€ข 4 problem set
โ€ข 21 instructional days
โ€ข Topic A: The Story of Trigonometry and Its Context
โ€ข Topic B: Understanding Trigonometric Functions and Putting
Them to Use
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Mathematical Themes of Module 2
โ€ข Emphasis on exploratory lessons rather than problem set
lessons
โ€ข Developing the sine and cosine functions through exploration
of the horizontal and vertical displacement of a car on a
rotating Ferris wheel
โ€ข Defining sine and cosine as functions of real numbers
โ€ข Graphing the the sine and cosine functions and
transformations of these functions
โ€ข Discovering and proving trigonometric identities
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
What are students coming in with?
โ€ข Experience with graphing transformations of functions on the
coordinate plane (G9)
โ€ข Experience modeling data using appropriate functions (G9 and
G11)
โ€ข Experience with trigonometric ratios and circles (G10)
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
โ€ข
โ€ข
โ€ข
โ€ข
โ€ข
Overview of Module 2
Topic A
Mid-Module Assessment
Topic B
End of Module Assessment
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Topic A: The Story of Trigonometry and Its
Contexts
โ€ข 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 .
โ€ข Students are given the historical context of how trigonometry
was developed.
โ€ข The values of the sine and cosine functions for multiples of , ,
and are used to generate the graphs of the sine and cosine
functions.
โ€ข The graphs of the sine and cosine function are used to discover
basic trigonometric identities.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 1 and 2: Ferris Wheels โ€“ Tracking
the Height and Co-Height of a Passenger
Car
โ€ข Students model the circular motion of a Ferris wheel using a
paper plate.
โ€ข Sine and cosine are not mentioned in these lessons.
โ€ข The functions are described as the โ€œheight and co-height
functions.โ€
โ€ข The lessons employ a familiar scenario to introduce the ideas
of circular motion and periodic functions.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 1 and 2: Ferris Wheels โ€“ 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 3: The Motion of the Moon, Sun,
and Starsโ€”Motivating Mathematics
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 4: From Circle-ometry to
Trigonometry
โ€ข
โ€ข
โ€ข
โ€ข
Work example 1 and 2.
Work problem set 2 โ€“ 5.
Lots of information to process in this lesson.
Consider using the scaffold suggestion of creating a graphic
organizer.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 4: From Circle-ometry to
Trigonometry
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A โ€œHandyโ€ Trick
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A โ€œHandyโ€ Trick
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 5: Extending the Domain of Sine
and Cosine to All Real Numbers
โ€ข Students explore clockwise rotations and rotations greater
than 360 degrees
โ€ข Students also explore rotations in which the terminal ray lands
on the axis or axis.
โ€ข Work through the discussion case 1 and 2.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 6: Why Call it Tangent?
โ€ข Work the opening exercise.
โ€ข You may need to review some terminology from geometry.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 6: Why Call it Tangent?
โ€ข Students interpret the tangent function several ways:
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 7: Secant and the Co-Functions
โ€ข What is the geometric meaning of the secant function?
โ€ข Why is cosecant the reciprocal of the sine function rather than the cosine
function?
โ€ข How do we make sense of the domain and range of the various
trigonometric functions?
โ€ข Do we really need these three trigonometric functions?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Sprint
Evaluating trigonometric functions
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 8: Graphing the Sine and Cosine
Functions
โ€ข 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.
โ€ข The emphasis should be on the key features of each graph.
โ€ข Relative maxima and minima
โ€ข Intervals of increasing and decreasing
โ€ข Intercepts
โ€ข End behavior
โ€ข Symmetry
โ€ข Amplitude
โ€ข Period
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 9: Awkward! Who Chose the
Number 360 Anyway?
โ€ข Why do we need radian measurement when degrees have worked fine up
to this point?
โ€ข Arc length calculations are easier for radian measurements.
โ€ข When graphing sine and cosine functions, the vertical and horizontal axes must be
scaled completely differently.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 9: Awkward! Who Chose the
Number 360 Anyway?
โ€ข The main reason that we see the transition from degrees to radians is that
all trigonometry done in Calculus is based on radian measurement.
โ€ข Find lim
x ๏‚ฎ0
sin( x)
.
x
โ€ข 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 9: Awkward! Who Chose the
Number 360 Anyway?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 9: Awkward! Who Chose the
Number 360 Anyway?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 10: Basic Trigonometric Identities
from Graphs
โ€ข Students use the graphs of the sine and cosine functions to note properties
that lead to trigonometric identities.
โ€ข Periodicity
โ€ข Symmetry (even/odd functions)
โ€ข Horizontal translations
โ€ข Note that an identity is a statement that two functions are equal on a
common domain.
โ€ข The statement โ€œsin(x+2ฯ€)=sin(x)โ€ itself is not an identity.
โ€ข The correct statement is โ€œsin(x+2ฯ€)=sin(x) for all real numbers x.โ€
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 10: Basic Trigonometric Identities
from Graphs
โ€ข Key takeaways from this lesson:
โ€ข The basic trigonometric identities derived from the periodic nature and symmetry
of the graphs of each function.
โ€ข For all real numbers x,
โ€ข The graphs of sine and cosine can be made to coincide with one another by using
horizontal shifts.
โ€ข For all real numbers x,
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Key Points โ€“ Topic A
โ€ข There is a rich historical context that motivates the study of
trigonometry.
โ€ข Students are given opportunities to explore the meaning of the
trigonometric functions and their properties in order to
develop a deeper understanding.
โ€ข 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
โ€ข
โ€ข
โ€ข
โ€ข
โ€ข
Overview of Module 2
Topic A
Mid-Module Assessment
Topic B
End of Module Assessment
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Mid-Module Assessment
Work with a partner on this assessment
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Scoring the Assessment
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
โ€ข
โ€ข
โ€ข
โ€ข
โ€ข
Overview of Module 2
Topic A
Mid-Module Assessment
Topic B
End of Module Assessment
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Topic B: Understanding Trigonometric
Functions and Putting Them to Use
โ€ข Students use trigonometric functions to model periodic behavior.
โ€ข The concept of polynomial identities covered in module 1 is extended to
trigonometric identities.
โ€ข 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 11: Transforming the Graph of the
Sine Function
โ€ข Students explore the family of functions given by the generalized sine
function .
โ€ข 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 11: Transforming the Graph of the
Sine Function
โ€ข For a sine function written in the form ๐‘“ ๐‘ฅ = ๐ด sin(๐œ”( ๐‘ฅ โˆ’ โ„Ž)) + ๐‘˜, for
real numbers ๐ด,๐œ”,โ„Ž, and ๐‘˜,
โ€ข |๐ด| is called the amplitude of the function.
โ€ข
2๐œ‹
|๐œ”|
is the period of the function.
โ€ข
|๐œ”|
2๐œ‹
is the frequency of the function (the frequency is the reciprocal of the period).
โ€ข โ„Ž is called the phase shift.
โ€ข the graph of ๐‘ฆ = ๐‘˜ is called the midline.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 12: Ferris Wheelsโ€”Using
Trigonometric Functions to Model Cyclical
Behavior
โ€ข Work through the exploratory challenge (exercises 1 โ€“ 5).
โ€ข Consider the height function from exercise 4:
โ€ข How would this formula change for a Ferris wheel with a different diameter?
โ€ข How would this formula change for a Ferris wheel at a different height off the
ground?
โ€ข How would this formula change for a Ferris wheel that had a different rate of
revolution?
โ€ข 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?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 13: Tides, Sound Waves, and Stock
Markets
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 14: Graphing the Tangent Function
โ€ข Students work in groups where each group is responsible for creating one
branch of the graph of the tangent function.
โ€ข These branches are then combined into a single graph.
โ€ข Students then work on trigonometric identities involving tangent.
โ€ข Continue to be precise when stating an identity.
sin(๐‘ฅ)
๐œ‹
โ€ข tan(๐‘ฅ) = cos(๐‘ฅ) for x โ‰  2 + ๐‘˜๐œ‹, for all integers ๐‘˜
โ€ข Note that is used when working on a circle and is used when working on a
coordinate plane.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 14: Graphing the Tangent Function
โ€ข At this point, it would be valuable to compare identities developed
throughout this module.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 15: What is a Trigonometric
Identity?
โ€ข The lesson opens by developing the
Pythagorean Identity.
โ€ข Since ๐‘ฅ๐œƒ = cos(๐œƒ) and ๐‘ฆ๐œƒ = sin(๐œƒ), it
follows that cos 2 ๐œƒ + sin2 ๐œƒ = 1.
โ€ข Students then derive a second identity
from the Pythagorean Theorem.
โ€ข
tan2 ๐œƒ + 1 = sec 2 ๐œƒ for all real numbers ๐œƒ
๐œ‹
such that ๐œƒ โ‰  + ๐‘˜๐œ‹, for all integers ๐‘˜.
2
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 16: Proving Trigonometric
Identities
โ€ข Is the following statement a trigonometric identity for all ๐œƒ?
sin(๐œƒ) + cos(๐œƒ) = โˆ’ 1 + 2sin(๐œƒ)cos(๐œƒ)
โ€ข What is wrong with this mathematical proof?
First, [1] sin ๐œƒ + cos ๐œƒ = โˆ’ 1 + 2 sin ๐œƒ cos ๐œƒ for ๐œƒ any real number.
Using the multiplication property of equality, square both sides, which gives
[2] sin2 ๐œƒ + 2sin ๐œƒ cos ๐œƒ + cos 2 ๐œƒ = 1 + 2sin ๐œƒ cos ๐œƒ for ๐œƒ any
real number.
Using the subtraction property of equality, subtract 2sin ๐œƒ cos ๐œƒ from each
side, which gives [3] sin2 ๐œƒ + cos 2 ๐œƒ = 1 for ๐œƒ any real number.
Statement [3] is the Pythagorean identity. So, replace sin2 ๐œƒ + cos 2 ๐œƒ by 1
to get [4] 1 = 1, which is definitely true.
Therefore, the original statement must be true.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 17: Trigonometric Identity Proofs
โ€ข Wishful thinking: By any chance, does sin ๐‘ฅ + ๐‘ฆ = sin ๐‘ฅ + sin(๐‘ฆ)?
โ€ข Students first find a formula sin ๐‘ฅ + ๐‘ฆ by exploring patterns (MP 8).
โ€ข They then prove their conjecture for two positive numbers whose sum is
๐œ‹
less than .
2
โ€ข Work Examples 1 and 2.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Key Points โ€“ Topic B
โ€ข Trigonometric functions are useful for modeling periodic data.
โ€ข A trigonometric identity is a statement that two functions are
equivalent on a given domain and an identification of that
domain.
โ€ข Trigonometric identities can be proven graphically, numerically, and
algebraically.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Key Points โ€“ Module 2 Lessons
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
โ€ข
โ€ข
โ€ข
โ€ข
โ€ข
Overview of Module 2
Topic A
Mid-Module Assessment
Topic B
End of Module Assessment
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment
Work with a partner on this assessment
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Scoring the Assessment
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Key Points โ€“ End-of-Module Assessment
โ€ข 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.
โ€ข Recall, as much as possible, assessment items are designed to
asses the standards while emulating PARCC Type 2 and Type 3
tasks.
โ€ข Recall, rubrics are designed to inform each district / school /
teacher as they make decisions about the use of assessments
in the assignment of grades.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
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?
© 2012 Common Core, Inc. All rights reserved. commoncore.org