Periodic Phenomena Unit Plan - justineheroux.weebly.comjustineheroux.weebly.com/uploads/3/1/8/4/31849211/edtpa_lesson_plans_for_learning... · Periodic Phenomena Unit Scope and Sequence

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)


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


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.


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.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.


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


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.


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.



SMART Board

Computer


Projector

Dry-erase board


Pens, pencils

Graphing calculator

Unit Arc plan


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.


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.






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.



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.




prompted at the end of the lesson to explore a piece of information that is important in the

following 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.




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.7 Read and listen for logical understanding of mathematical thinking shared by

other students.




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.


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.


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).


SMART Board Lesson 3 Presentation (please see Key Instructional Materials)

SMART Board

Computer

Projector

Dry-erase board



Pens, pencils

Graphing or scientific calculator

Unit Arc plan


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


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.







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.



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.



prompted at the end of the lesson to explore a piece of information that is important in the

following lesson.


Appendix:







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


including numerical tables, formulas, functions, equations, charts, graphs, and diagrams




mathematical idea.

A2.CN.2 Understand the corresponding procedures for similar problems or mathematical

concepts.


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.


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).



SMART Board

Computer

Projector

Dry-erase board


Pens, pencils

Graphing calculator


Unit Arc plan


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


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)






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.



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.


Appendix:






Documents

Periodic Phenomena Unit Plan - justineheroux.weebly.comjustineheroux.weebly.com/uploads/3/1/8/4/31849211/edtpa_lesson_plans_for_learning... · Periodic Phenomena Unit Scope and Sequence