periodic phenomena unit plan -...
TRANSCRIPT
Periodic Phenomena Unit Scope and Sequence
Periodic Phenomena Unit Plan
Monday
Tuesday
Wednesday
Thursday
Friday
Sat/Sun
February 24 February 25 February 26 February 27 February 28 1/2
Group V- Poly
V- Poly
V- Poly
V- Poly
Group U- Go over new unit arc and take diagnostic exam
U- Lesson 1: Periodic Phenomena Connecting the Unit Circle to sine and cosine graphs
U- Lesson 2: Describing Periodic Phenomena: Period and Frequency
X- Lesson 2: Describing Periodic Phenomena: Period and Frequency
X- Lesson 3: Describing Periodic Phenomena: Amplitude, Frequency, Period, and Phase Shifts
Group X- Go over new unit arc and take diagnostic exam
X- Lesson 1: Periodic Phenomena Connecting the Unit Circle to sine and cosine graphs
U- Cont. X- Cont. U- Lesson 3: Describing Periodic Phenomena: Amplitude, Frequency, Period, and Phase Shifts
March 3 March 4 March 5 March 6 March 7 Group V- Poly
V- Poly
V- Poly
V- Poly
Group U- Lesson 4: Sketching and recognizing y=tan(x), y=csc(x), y=cot(x), and y=sec(x)
U- Independent work time and Responsibility group Learning Conferences
U- Independent work time and Responsibility group Learning Conferences
X- Independent work time and Responsibility group Learning Conferences
X- Unit Quiz
Group X- Lesson 4: Sketching and recognizing y=tan(x), y=csc(x), y=cot(x), and y=sec(x)
X- Independent work time and Responsibility group Learning Conferences
U- Cont. X- Cont. U- Unit Quiz
Heroux, Lesson 1 Page 1
Lesson 1 Outline
Teacher: Ms. Justine Heroux Date: February 25, 2014
Course: Algebra 2 Trigonometry Topic: Periodic Functions, Sine and Cosine
Learning Standards
Process Strands
A2.CM.2 Use mathematical representations to communicate with appropriate accuracy,
including numerical tables, formulas, functions, equations, charts, graphs, and diagrams.
A2.CN.1 Understand and make connections among multiple representations of the same
mathematical idea.
A2.R.2 Recognize, compare, and use an array of representational forms.
A2.R.3 Use representation as a tool for exploring and understanding mathematical ideas.
A2.R.6 Use mathematics to show and understand physical phenomena (e.g., investigate
sound waves using the sine and cosine functions).
A2.RP.4 Recognize when an approximation is more appropriate than an exact answer.
Content Strand
A2.A.70 Sketch and recognize one cycle of a function of the form y=AsinBx or
y=AcosBx.
Objectives: The students will use their past knowledge of trigonometric functions and the Unit
Circle to help sketch the periodic sine and cosine curves y=sin(x) and y=cos(x). We will discuss
what it means to be periodic, where we see periodic phenomena in nature, and how to sketch the
sine and cosine curves. Afterwards the students will use the lesson and classroom resources for
independent and group work to complete the required materials listed on their learning contract
(Unit Arc).
Prerequisite skills: Students should be able to recall and use the Unit Circle, including the
coordinates, degrees, and equivalent radians.
A2.A.56 Know the exact and approximate values of the sine, cosine, and tangent of 0º,
30º, 45º, 60º, 90º, 180º, and 270º angles.
A2.A.66 Determine the trigonometric functions of any angle, using technology.
Textbook/Materials/Equipment:
SMART Board Lesson 1 Presentation (please see Instructional Materials)
SMART Board
Computer
Projector
Dry-erase board
Hand-made “Paper Plate Trig” manipulative from the previous unit (please see Key
Instructional Materials)
Periodic functions note packet (please see Assessments)
Pens, pencils
Graphing or scientific calculator
Unit Arc plan
Books and worksheets referenced on the Unit Arc (please see Instructional Materials)
Heroux, Lesson 1 Page 2
Teaching strategies
Overall: Some direct teaching with visual aids, interactive SMART board widgets,
technology, and manipulatives. Learning cultures, responsibility groups, and unison reading
created by Cynthia McCalister and implemented by the school also drives the lessons and
curriculum. This lesson is intended as a scaffold for the lessons to follow.
Diverse Needs Students: Visual aids, the Unit Circle manipulative, whole class
instruction, and chosen independent work will help access the different learning styles,
intelligences, and academic levels of the students. The purpose of the independent work in each
class is to aid in the development of greater executive functioning and college ready skills. While
responsibility groups are meant as a motivator to complete given tasks, share common
experiences, and gain knowledge.
Lesson
Description/Format: Lesson is from 12:55pm-1:46pm
4-6 minutes: Attendance, preparation of materials, return Diagnostic Exam from the
previous day
15-25 minutes: Presentation on Periodic Phenomena
2-3 min: What does it mean to be periodic?
2-3 min: Whole class discussion on where we see periodicity in nature
2-5 min: Review of the unit circle, use hand-made unit circle manipulative, and
develop the connection to periodic functions
10-14 min: Use the unit circle to draw the periodic functions y=sin(x) and
y=cos(x) by creating a coordinate chart for each function, graphing the points,
then sketching the curve.
15-20 minutes: Independent/responsibility group work time. Students can work on book
work listed on the Unit Arc or any of the practice quizzes.
Motivation: Students will be able to share their experiences with periodic phenomena and see or
hear about some interesting ones they may not have previously known. This will lead into the
lesson and unit topic; how we can describe periodic phenomena through trigonometric functions.
I will also use previous knowledge and hooks as a motivation. Last unit, the students completed a
“Paper Plate Trig” manipulative to help develop their knowledge of the unit circle and its
properties. This lesson the students can translate how they found “Bob’s location” last unit (by
using their manipulative and the coordinates on the Unit Circle) to the activity in this unit on
graphing the Sine and Cosine functions.
Board presentation/Overheads/Handouts:
Periodic Functions Packet Lesson 1 (please see Assessments)
Lesson 1 SMART presentation (please see Key Instructional Materials)
Unit Arc plan (please see Key Instructional Materials)
Learning Target Practice Quizzes 1-4 (please see Assessments)
Diagnostic Exam (please see Assessments)
Critical thinking questions: What does it mean to be periodic? Where do we see periodic
phenomena in the real world? How can we use trigonometry to represent and describe periodic
Heroux, Lesson 1 Page 3
phenomena? How can we use our previous knowledge of the unit circle to develop and describe
these periodic functions? Why can we use the Unit Circle to form the graphs of y=sin(x) and
y=cos(x)?
Connections: Students will make connections between the periodic phenomena they see in real
life, such as the seasons, to mathematics. Students will then connect their knowledge of the
previous units to their work in this unit during the lesson activities. Students will also use the
sine and cosine functions in future lessons relating to amplitude, period, frequency, and phase
shifts in order to better describe periodic phenomena.
Homework: To finish the note pages for the Lesson 1 (a required assignment) and start their self-
check assignments. Please see attached Unit Arc in Key Instructional Materials for list of
assignments.
Assessment: During the lesson I will assess the students on their attentiveness and participation.
Poor participation lets me know who to make sure to see during independent work time and who
to ask to stay after school. The students also have a list of required and choice self-check
activities, all of which are in preparation for their unit quiz on the following Friday. For instance,
the Periodic Functions packet will be collected and graded at the end of the week. This is one
way for me to assess how well they followed along with each lesson. Please see attached Unit
Arc for details.
Extension: Students are able to use their independent work time and listed resources in the Unit
Arc to explore an area they do not understand or want to learn more about. The students are also
prompted at the end of their work packet to explore a piece of information that is important in the
following lesson.
Reflection: To be completed after the lesson.
Appendix:
When I present the lesson to the class the graphs, charts, and questions will be blank. I
then will fill in the slides as we complete the lesson. Students can then refer to the
completed presentation, past presentations, and unit as a whole on the class webpage.
Images in the presentation and note packets are cited at the bottom of each page of the
SMART document and word documents.
Heroux, Lesson 2 Page 1
Lesson 2 Outline
Teacher: Ms. Justine Heroux Date: February 26th
and 27th
, 2014
Course: Algebra 2 Trigonometry Topic: Describing Periodic Phenomena,
Changing the Period and Frequency of Sine
and Cosine
Learning Standards
Process Strands
A2.CM.1 Communicate verbally and in writing a correct, complete, coherent, and clear
design (outline) and explanation for the steps used in solving a problem.
A2.CM.2 Use mathematical representations to communicate with appropriate accuracy,
including numerical tables, formulas, functions, equations, charts, graphs, and diagrams.
A2.CM.7 Read and listen for logical understanding of mathematical thinking shared by
other students.
A2.CM.12 Understand and use appropriate language, representations, and terminology
when describing objects, relationships, mathematical solutions, and rationale.
A2.CN.1 Understand and make connections among multiple representations of the same
mathematical idea.
A2.PS.3 Observe and explain patterns to formulate generalizations and conjectures.
A2.PS.7 Work in collaboration with others to propose, critique, evaluate, and value
alternative approaches to problem solving.
A2.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or
objects created using technology as representations of mathematical concepts
A2.R.6 Use mathematics to show and understand physical phenomena (e.g., investigate
sound waves using the sine and cosine functions).
Content Strand
A2.A.69 Determine amplitude, period, frequency, and phase shift, given the graph or
equation of a periodic function.
A2.A.70 Sketch and recognize one cycle of a function of the form y=AsinBx or
y=AcosBx.
Objectives: By the end of the lesson students will be able to determine the period and frequency
of a given function in form y=AsinBx or y=AcosBx. They will then be able to sketch a curve of
the function in the form of y=AsinBx or y=AcosBx from 0 to 2π when A is 1 and B changes. In
order to sketch the functions the students will call upon the characteristics of the sinusoidal
functions sine and cosine as well as determine their period and frequency.
Prerequisite skills: Students should be able to recognize one cycle of y=sin(x) and y=cos(x).
They should also be able to recall the definitions for period and frequency, and what it means for
a function to be periodic.
Textbook/Materials/Equipment:
SMART Board Lesson 2 Presentation (please see Instructional Materials)
SMART Board
Computer
Heroux, Lesson 2 Page 2
Projector
Dry-erase board
Periodic functions note packet (please see Assessments)
Pens, pencils
Graphing calculator
Unit Arc plan
Books and worksheets referenced on the Unit Arc (please see Instructional Materials)
Teaching strategies
Overall: Some direct instruction with visual aids, interactive SMART board widgets, and
graphing technology. Learning cultures, responsibility groups, and unison reading also drives the
lessons and curriculum. This lesson is the next scaffold in place for the students to be able to
choose trigonometric functions to model and describe periodic phenomena with specified
amplitude, frequency, and midline.
Diverse Needs Students: Visual aids, corresponding notes, whole class instruction, and
chosen independent work will help access the different learning styles, intelligences, and
academic levels of the students. The purpose of the independent work in each class is to aid in
the development of greater executive functioning and college ready skills. While responsibility
groups are meant as a motivator to complete given tasks, share common experiences, and gain
knowledge.
Lesson
Description/Format: Lesson is from 12:01-1:46 pm
4-5 minutes: Attendance, class discussion and review of y=cos(x) and y=sin(x) graph
characteristics.
30-40 minutes: Class practice and period and frequency exploration.
3-5 min: Compare different periodic phenomena when graphed. Ask “What do you
notice about the curves?”. Hold a discussion on the differences between the graphs.
Recall the definition of period and frequency.
3-5 min: Exploration using the SMART board trigonometric functions widget what
changes in the sinusoidal curves when the coefficient B changes in y=cos(Bx) and
y=sin(Bx). As the coefficient B increases, frequency increases, and period
decreases. As the coefficient B decreases to a smaller and smaller fraction,
frequency decreases, and period increases.
3-5 min: Using variables, define the period and frequency of sine and cosine in
y=cos(Bx) and y=sin(Bx).
4-6 min: Whole class practice finding the period and frequency of the periodic
functions sine and cosine when given the function.
3-5 min: Overview and practice on how to graph and sketch a function in the form
y=Asin(Bx) and y=Acos(Bx) using a graphing calculator. Students will write an
outline of the steps for Method 1 in their note packets.
3-5 min: Overview and practice on how to graph and sketch a function in the form
y=Asin(Bx) and y=Acos(Bx) without using a graphing calculator. Students will
write an outline of the steps for Method 2 in their note packets.
5-10 min: Whole class practice sketching two of the four given functions from 0 to
2π, the rest of the questions are independent practice.
Heroux, Lesson 2 Page 3
55-60 minutes: I will have 10-15 minute learning conferences with each of the
responsibility groups to determine what they have accomplished as a group and as
individuals. We will then discuss what concepts the students are still struggling with, and
review those concepts. The rest of the class will have the choice to work independently or
with their responsibility groups. During this time students can work on finishing the
packet practice problems, the book work listed on the Unit Arc, or any of the practice
quizzes.
Motivation: Students will refer back to the last lesson to determine easy characteristics of
y=sin(x) and y=cos(x) to help them differentiate the two (waves versus a bowl). They will then
be able to see and explore how frequency changes a periodic function by hearing and imagining
high and low pitches, as well as seeing the graph change on the SMART widget. Students will
be able to come up to the SMART board, type in their own coefficient B values, and visually see
how the graph changes. I will also demonstrate high and low pitch sounds.
Board presentation/Overheads/Handouts:
Periodic Functions Packet Lesson 2 (please see Assessments)
Lesson 2 SMART presentation (please see Key Instructional Materials)
Unit Arc plan (please see Key Instructional Materials)
Learning Target Practice Quizzes 1-4 (please see Assessments)
Critical thinking questions: How do the graphs y=sin(x) and y=cos(x) differ? What
characteristics do we notice or think about when we see a graph of y=sin(x)? How about
y=cos(x)? How does light and sound change the period and frequency of a curve? Do all periodic
functions appear the same when graphed? What happens when we change the coefficient B in
our functions? Why? How does the coefficient B change the period and frequency of y=sin(Bx)
and y=cos(Bx)? What is one possible method you can use to sketch y=Asin(Bx) and
y=Acos(Bx)?
Connections: Students will connect trigonometric curves to instrument sounds and star light.
Connecting concrete phenomena to abstract curves will foster their ability to describe periodic
phenomena.
Homework: To finish the note pages for the Lesson 2 (a required assignment) and start their self-
check assignments. Please see attached Unit Arc in Key Instructional Materials for list of
assignments.
Assessment: During the lesson I will assess the students on their attentiveness and participation
throughout the lesson, independent work time, and our learning conferences. Learning
conferences are documented on a graphic organizer to show when students participate and what
they say. Poor participation lets me know who to make sure to see during independent work
time, talk to during the conference, or to ask to stay after school. The students will also be
assessed on their self-check activities and the Periodic Functions packet.
Extension: Students are able to use their independent work time and listed resources in the Unit
Arc to explore an area they do not understand or want to learn more about. The students are also
Heroux, Lesson 2 Page 4
prompted at the end of the lesson to explore a piece of information that is important in the
following lesson.
Reflection: To be completed after the lesson.
Appendix:
When I present the lesson to the class the graphs, charts, and questions will be blank. I then
will fill in the slides as we complete the lesson. Students can then refer to the completed
presentation, past presentations, and unit as a whole on the class webpage.
Images in the presentation and note packets are cited at the bottom of each page of the
SMART document and word documents.
Heroux, Lesson 3 Page 1
Lesson 3 Outline
Teacher: Ms. Justine Heroux Date: February 28, 2014
Course: Algebra 2 Trigonometry Topic: Describing Periodic Phenomena,
determining the Period, Amplitude,
Frequency, and Phase Shifts of Sine and
Cosine
Learning Standards
Process Strands
A2.CM.2 Use mathematical representations to communicate with appropriate accuracy,
including numerical tables, formulas, functions, equations, charts, graphs, and diagrams.
A2.CM.7 Read and listen for logical understanding of mathematical thinking shared by
other students.
A2.CM.12 Understand and use appropriate language, representations, and terminology
when describing objects, relationships, mathematical solutions, and rationale.
A2.CN.1 Understand and make connections among multiple representations of the same
mathematical idea.
A2.PS.3 Observe and explain patterns to formulate generalizations and conjectures.
A2.R.2 Recognize, compare, and use an array of representational forms.
A2.R.3 Use representation as a tool for exploring and understanding mathematical ideas.
A2.R.6 Use mathematics to show and understand physical phenomena (e.g., investigate
sound waves using the sine and cosine functions)
Content Strand
A2.A.69 Determine amplitude, period, frequency, and phase shift, given the graph or
equation of a periodic function.
A2.A.70 Sketch and recognize one cycle of a function of the form y=AsinBx or
y=AcosBx.
A2.A.72 Write the trigonometric function that is represented by a given periodic graph.
Objectives: Students will be able to determine the amplitude, period, frequency, midline, vertical
and horizontal phase shifts in a given periodic graph and periodic function. They can then use
those characteristics to either determine the function when given a curve, or sketch the curve
when given a function.
Prerequisite skills: When provided with a graph, students should be able to recognize if the
function is sine or cosine. They should also be able to determine and write the period and
frequency of a given graph. Then if given a function, students should be able to sketch and
recognize at least one cycle of y=sin(Bx) and y=cos(Bx).
Textbook/Materials/Equipment:
SMART Board Lesson 3 Presentation (please see Key Instructional Materials)
SMART Board
Computer
Projector
Dry-erase board
Heroux, Lesson 3 Page 2
Periodic functions note packet (please see Assessments)
Pens, pencils
Graphing or scientific calculator
Unit Arc plan
Books and worksheets referenced on the Unit Arc (please see Instructional Materials)
Teaching strategies
Overall: Some direct teaching with visual aids, interactive SMART board widgets,
corresponding notes, and technology. Learning cultures, responsibility groups, and unison
reading also drives the lessons and curriculum. This lesson is intended as a scaffold for the
lessons to follow.
Diverse Needs Students: Visual aids, whole class instruction, and chosen independent
work will help access the different learning styles, intelligences, and academic levels of the
students. The purpose of the independent work in each class is to aid in the development of
greater executive functioning and college ready skills. While responsibility groups are meant as a
motivator to complete given tasks, share common experiences, and gain knowledge.
Lesson
Description/Format: Lesson is from 12:01pm-12:53pm
3-5 minutes: Attendance, review on how to find period and frequency in a given function.
15-35 minutes: Changing the coefficients A, B, C, D in y=Acos(B(x-C))+D and
y=Asin(B(x-C))+D exploration.
4-5 min: Use SMART widget to visually see what part of the curve changes in the
graphs of sine and cosine when coefficients A, C, and D are each changed.
Review and compare to when the coefficient B is changed.
5-8 min: Define amplitude, midline, horizontal and vertical phase shifts while
matching the mathematical terms to their respective coefficient.
8-12 min: Whole class practice identifying amplitude, period, frequency, midline,
vertical and horizontal phase shifts when given a periodic function.
8-10 min: Whole class practice determining the periodic function when given a
curve by identifying the period, amplitude, midline, frequency, vertical and
horizontal phase shifts.
8-13 minutes: Independent/responsibility group work time. Students can work on book
work listed on the Unit Arc or any of the practice quizzes.
Motivation: Students will be able to use the SMART board to change given coefficients and
share with the class their opinion on how the curve changed. This activity will be similar to the
last lesson but with more coefficients. So students can use their previous knowledge to predict
and then interpret the change in the curves. The goal is to incite problem solving, debate,
exploration, and discovery.
Board presentation/Overheads/Handouts:
Periodic Functions Packet Lesson 3 (please see Assessments)
Lesson 3 SMART presentation (please see Key Instructional Materials)
Unit Arc plan (please see Key Instructional Materials)
Learning Target Practice Quizzes 1-4 (please see Assessments)
Heroux, Lesson 3 Page 3
Critical thinking questions: When we change coefficient A, what happens to our sine curve?
Does the same change occur with the cosine curve? Does the change look similar to when we
change coefficient B? What is different? How about the coefficients C and D? How do you
determine the period, amplitude, frequency, midline, vertical and horizontal phase shift when
given a function? How can we use those characteristics to draw a curve? How do you determine
the period, amplitude, frequency, midline, vertical and horizontal phase shift when given a
curve? How can we use those characteristics to write a function?
Connections: Students will be able to make connections between the previous lessons and this
lesson on how coefficients in a trigonometric function change the drawn curve. They will
compare the changed curves to what they know to be true for y=sin(x) and y=cos(x). This will
continue the idea of accurately describing periodic phenomena.
Homework: To finish the note pages for the Lesson 3 (a required assignment) and start their self-
check assignments. Please see attached Unit Arc in Key Instructional Materials for list of
assignments.
Assessment: Throughout the lesson I will assess the students on their attentiveness and
participation during lesson and in group/independent work time. Poor participation lets me know
who to make sure to see during independent work time and who to ask to stay after school. The
students will also be assessed on their self-check activities and the Periodic Functions packet.
Extension: Students are able to use their independent work time and listed resources in the Unit
Arc to explore an area they do not understand or want to learn more about. The students are also
prompted at the end of the lesson to explore a piece of information that is important in the
following lesson.
Reflection: To be completed after the lesson.
Appendix:
When I present the lesson to the class the graphs, charts, and questions will be blank. I
then will fill in the slides as we complete the lesson. Students can then refer to the
completed presentation, past presentations, and unit as a whole on the class webpage.
Images in the presentation and note packets are cited at the bottom of each page of the
SMART document and word documents.
Heroux, Lesson 4 Page 1
Lesson 4 Outline
Teacher: Ms. Justine Heroux Date: March 3, 2014
Course: Algebra 2 Trigonometry Topic: Sketching and Recognizing y=tan(x),
y=cot(x), y=csc(x), and y=sec(x).
Learning Standards
Process Strands
A2.CM.2 Use mathematical representations to communicate with appropriate accuracy,
including numerical tables, formulas, functions, equations, charts, graphs, and diagrams
A2.CM.12 Understand and use appropriate language, representations, and terminology
when describing objects, relationships, mathematical solutions, and rationale.
A2.CN.1 Understand and make connections among multiple representations of the same
mathematical idea.
A2.CN.2 Understand the corresponding procedures for similar problems or mathematical
concepts.
A2.PS.3 Observe and explain patterns to formulate generalizations and conjectures.
A2.PS.8 Determine information required to solve the problem, choose methods for
obtaining the information, and define parameters for acceptable solutions.
A2.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or
objects created using technology as representations of mathematical concepts.
A2.R.6 Use mathematics to show and understand physical phenomena (e.g., investigate
sound waves using the sine and cosine functions)
Content Strand
A2.A.71 Sketch and recognize the graphs of the functions y=sec(x), y=csc(x), y=tan(x),
and y=cot(x).
Objectives: Students will be able to sketch and recognize the graphs of the functions y=sec(x),
y=csc(x), y=tan(x), and y=cot(x). Students will explore what it means for these functions to be
undefined and the purpose of an asymptote on a graph.
Prerequisite skills: Students will need to recall the six trigonometric functions and their ratios for
sine, cosine, tangent, cotangent, cosecant, and secant. It is also important to be able to recall what
it means to divide a number, and possibly a function, by zero. Students will also use their
knowledge of the unit circle and the graphs of y=sin(x) and y=cos(x) to help sketch and
recognize functions y=sec(x), y=csc(x), y=tan(x), and y=cot(x).
Textbook/Materials/Equipment:
SMART Board Lesson 4 Presentation (please see Instructional Materials)
SMART Board
Computer
Projector
Dry-erase board
Periodic functions note packet (please see Assessments)
Pens, pencils
Graphing calculator
Heroux, Lesson 4 Page 2
Unit Arc plan
Books and worksheets referenced on the Unit Arc (please see Instructional Materials)
Teaching strategies
Overall: Some direct teaching with visual aids and corresponding notes. Learning
cultures, responsibility groups, and unison reading drives the lessons and curriculum. This lesson
is the final new material lesson of the unit.
Diverse Needs Students: Visual aids, whole class instruction, and chosen independent
work will help access the different learning styles, intelligences, and academic levels of the
students. The purpose of the independent work in each class is to aid in the development of
greater executive functioning and college ready skills. While responsibility groups are meant as a
motivator to complete given tasks, share common experiences, and gain knowledge.
Lesson
Description/Format: Lesson is from 12:01pm-12:53pm
4-6 minutes: Attendance, review of the six trigonometric functions, their ratios, and their
relationship to sine and cosine.
30-47 minutes: Sketching y=tan(x), y=cot(x), y=csc(x), and y=sec(x)
8-10 min: Define where y=tan(x) is defined and undefined using the ratio
tan(x)=sin(x)/cos(x). Whole class discussion on if we can divide by zero and what
it means if the denominator of a function is zero both graphically and
algebraically. We will then sketch the graph of tangent as a class using those
asymptotes and a few nice coordinates from our unit circle such as the ones
defined at the radian measure 0, π/4, π/2, π, 3π/2, and 2π. We will also use pre-
drawn sine and cosine curves to help us graph these known points.
8-10 min: Define where y=cot(x) is defined and undefined using the ratio
cot(x)=cos(x)/sin(x) and the knowledge y=cot(x) is the inverse of y=tan(x). Again
we will use pre-drawn sine and cosine curves to help us graph known points.
8-10 min: Define where y=csc(x) is defined and undefined using the ratio
csc(x)=1/sin(x). We will use a pre-drawn sine curve to help us graph known points.
8-10 min: Define where y=sec(x) is defined and undefined using the ratio
sec(x)=1/cos(x). We will use a pre-drawn cosine curve to help us graph known
points.
5-7 min: Practice recognizing y=tan(x), y=cot(x), y=csc(x), and y=sec(x) when given the
graph. Then close the lesson by reviewing what is expected of the students on Friday.
Motivation: Students will be able to see another use for the six trigonometric ratios they used in
a past unit on right triangles. We will also test and debate if one can divide by zero in
mathematics using scaffolded questions. For example, we still start by dividing 4 by 4, then 4 by
2, then 4 by 1, and finally 4 by zero.
Board presentation/Overheads/Handouts: (please see attached)
Periodic Functions Packet Lesson 4 (please see Assessments)
Lesson 4 SMART presentation (please see Key Instructional Materials)
Unit Arc plan (please see Key Instructional Materials)
Learning Target Practice Quizzes 1-4 (please see Assessments)
Heroux, Lesson 4 Page 3
Critical thinking questions: How can we use our right triangle ratios to create a relationship
between sine, cosine, tangent, and cotangent? Can we divide by zero, why or why not? How can
we use tangent’s relationship to sine and cosine to sketch y=tan(x)? When is y=tan(x) undefined
and why? How can we use cotangent’s relationship to sine, cosine, and tangent to sketch
y=cot(x)? When is y=cot(x) undefined and why? How and why can we use the graph of y=sin(x)
to graph y=csc(x)? How and why can we use the graph of y=cos(x) to graph y=sec(x)? How can
we differentiate the graphs of the functions y=sec(x), y=csc(x), y=tan(x), and y=cot(x)?
Connections: Students will connect three different unit concepts to complete this lesson; their
unit on trigonometric ratios and right triangles, their unit on the Unit Circle, and the current unit
on graphing periodic functions. They will be able to use what they know about y=sin(x),
y=cos(x), unit circle coordinates, and trigonometric ratios to sketch defined and undefined areas
for the periodic functions y=sec(x), y=csc(x), y=tan(x), and y=cot(x). These characteristics and
connections will then enable the students to find differences between the four graphs so they can
recognize which graph corresponds to the four given functions.
Homework: To finish the note pages for the Lesson 4 (a required assignment) and start their self-
check assignments. Please see attached Unit Arc in Key Instructional Materials for list of
assignments.
Assessment: During the lesson I will assess the students on their attentiveness and participation.
Poor participation lets me know who to make sure to see during independent work time, or who
to ask to stay after school. The students will also be assessed on their self-check activities and the
Periodic Functions packet.
Extension: Students who understand how to sketch these functions will be able to help and
explain the graphs to other students next to them. In addition the students can sketch these graphs
on their own by completing practice quiz #3. The students can also continue to work on any of
their required or self-check work.
Reflection: To be completed after the lesson.
Appendix:
When I present the lesson to the class the graphs, charts, and questions will be blank. I
then will fill in the slides as we complete the lesson. Students can then refer to the
completed presentation, past presentations, and unit as a whole on the class webpage.
Images in the presentation and note packets are cited at the bottom of each page of the
SMART document and word documents.