**LESSON 11 (Approximately 2 hours; two weeks after lesson 10)**

**TRIGONOMETRY ‘CAROUSEL’**

*Overview*- Exploring trigonometric principles through instructional videos, interactive simulations, physical manipulatives, low-stakes games, conceptual maps, audio recordings, and multimedia ad campaigns
- Introducing applications for trigonometry including aviation, music, oceanography, and build projects
- Providing a ‘carousel’ of activities for students to explore in hopes of clarifying and extending students’ basic trigonometric toolkit
- Serving visual-spatial, bodily-kinesthetic, interpersonal, intrapersonal, linguistic, musical, and logical-mathematical learners

*Materials*3-12 ft lengths of rope

Student and teacher laptops or tablets with station websites already open

Class set of headphones

Voice recorders

Voice recording software (e.g., ‘VoiceThread’)

Colored pens and pencils

Dictionaries (4)

Interactive Simulators (e.g., ‘Netlogo’ and ‘PhET’)

*Key Words*Mathematical carousel | Trigonometry | Trigonometric functions | Unit circle | Sine | Cosine | Tangent | Cosecant | Secant | Cotangent | Inverse trigonometric functions | Arcsine | Arccosine | Arctangent | Arccotangent | Transformations | Oscillation | Ruben’s tube | Superposition | Interference patterns

__TEACHING PLAN__

*Introduction*Good morning! This lesson is meant to engage you—a wide array of learners with different tastes and preferences--through videos, interactive simulations, and games that dive into trigonometry, its many applications, and some of your varied interests. We will quickly revisit some of our earlier work on right triangle trigonometry so we can move onto the juicier stuff! From a bird’s eye perspective, we will begin learning about trigonometric functions, the unit circle, and properties of waves before investigating their applications in aviation and music. Today’s class is a carousel because you will be moving throughout the period from station to station. You might also consider it a sort of buffet. Please try everything out. Naturally, you will enjoy some things more than others and that is great; we expect that! Before you leave here today, you will hear about some options you have as far as the focus and deliverables for homework. Do you have any questions? I am really excited to dig into this with you!

__ASK Questions and discuss as a class__(15 minutes)

- What are the basic principles of sine, cosine and tangent? (
*At this juncture, compose essential notes on special right triangles, sine, cosine, & tangent as well as ‘Soh Cah Toa’*) - In what instances and for what problems might we use trigonometric functions versus inverse trigonometric functions? (
*With student input, present cogent word problems and diagrams that would call for the use of either trigonometric functions or inverse trigonometric functions*).

*Task*- (
*Distribute “Carousel Station Descriptions”)*Now, onto the carousel portion of the lesson. We will have eight stations set up around the room with computers at some of them. Four of you willtogether through these stations (like a carousel) while I circle the room helping wherever I can. You will have approximately eight minutes at each station. Here are the stations with quick descriptions, also available in the handout I just provided you: (5 minutes)*rotate*- Station 1: Here you will work individually on computers and
three holes of “Trig Mini Golf” http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID2=AB.MATH.JR.SHAP.TRIG&lesson=html/object_interactives/trigonometry/use_it.html*play* - Station 2: Using headphones,
a video entitled “Exploring Trigonometry (Video Interactive)”. Afterwards, with a partner,*watch*down two main points from the video. Then*write*why the ratio between altitude and horizontal distance must remain constant if the angle of descent is constant (i.e. three degrees with respect to the horizon). Finally*explain*with the other pairing at your station,*speak*your two main points and your explanation about the constant ratio. http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID2=AB.MATH.JR.SHAP.TRIG&lesson=html/video_interactives/trigonometry/trigonometrySmall.html*sharing* - Station 3: At this station you will explore the unit circle at your own pace and
60-90 seconds of audio on your initial observations and questions regarding…*record*- the relationships between sine and cosine waves
- radian versus degree measures and
- the relevance of special right triangles as they relate to sine, cosine, and tangent waves https://phet.colorado.edu/sims/html/trig-tour/latest/trig-tour_en.html

- Station 4: Here is a station where you’ll have the opportunity to blend your understanding of sine, cosine, and tangent with wave functions.
three diagrams from any three angles. After talking to a new partner from your quad at this station,*Draw*3-4 observations below your drawings https://www.mathsisfun.com/algebra/trig-interactive-unit-circle.html*bullet point* - Station 5: Here, you will
(i.e. ‘transform’) each of the three trigonometric functions discussed thus far: sine, cosine, and tangent. See what happens to the amplitude, period, and frequency of waves as you change values within functions. For now, focus primarily on sine, cosine, and tangent. https://illuminations.nctm.org/Activity.aspx?id=3589*manipulate* - Station 6: At Station 5, you learned how to transform functions by changing portions of functions. Here you will use a length of rope to try to
waves in a corner of the classroom. Please keep it cool in here but have fun*generate*frequency and period after quickly defining ‘transverse wave’, ‘frequency’, ‘wavelength’, and ‘amplitude’ with the dictionaries on the desk. Take turns*adjusting*transverse waves with the twelve-foot segment of rope provided in class. See if you can set up a constant frequency and amplitude. Expect a shoulder workout for sure!*modifying* - Station 7: This station has some sound effects so make sure the volume is up on the computers.
- First,
on the basic trigonometric identities while selecting your level of difficulty and mode (either degrees or radians): http://www.kwarp.com/portfolio/trigspinner.html*quiz yourself* - Second,
transverse, longitudinal, and periodic transverse waves: http://www.physics.nyu.edu/~ts2/Animation/Trans_Long_Periodic_Waves.html*compare & contrast* - Last,
to this question: Which waves do sine and cosine functions represent?*respond*

- First,
- Station 8:
minutes 0:00 though 4:50 of a video synthesizing several trigonometric concepts,*Watch*and pausing whenever it seems useful: https://www.youtube.com/watch?v=ovLbCvq7FNA and then watch minutes 4:50 through 6:30 and do the same*taking notes*

- Station 1: Here you will work individually on computers and
- So, we are about to start the carousel. I will let you know when time is up and when to switch stations. If each of you would please find yourself at any one of the labeled stations around the room that would be great. Please
yourselves so there are maximum four of you at each station. You only need to bring the “Carousel Station Description” handout with you and your math notebook. Let’s*arrange**circle*around the room (from a view above the classroom). (2 minutes)*clockwise* - (
*Students engage in Carousel activity*) (68 minutes) - Alright, let’s take a quick
break or*stretch*some trash can basketball shots. (2 minutes)*play*

*Wrap Up*__Discuss__

- I’d love to hear what you learned from the activity! (
*Introduce a ball toss activity during which time students throw a soft ball underhand to one another and share one or two observations they gleaned from ‘Carousel’*) (10 minutes) - Take four minutes to write a quick ‘K-W-L’ (Know-Want to Know-Learned) that you would not mind sharing. (4 minutes)
- Now, please crumple your K.W.L. up into a ball and throw it up into the air! (1 minute)
- Please take a moment to read somebody else’s K-W-L. before returning it to its rightful owner. (2 minutes).
- I would like each of you to now add something to your K-W-L. It could be something you just read or something that came to you in the last few minutes. After you have done this, please submit your K-W-L to me, making sure your name is on it. (3 minutes)
- Review homework deliverables due next class: (5 minutes)
__PART 1__:**Within Part 1 you have three options:****Option 1**: Learn more about sound wave interference and how speakers work- Play around with an interactive simulator that models two speakers interfering with one another. Technically, it should be possible to play sounds from two speakers and have one cancel the sound of the other out at a particular position in the room.
- You will download this link: http://ccl.northwestern.edu/netlogo/models/Speakers).
- Homework:
*Either*through one page of writing, a conceptual map with 3-4 diagrams,*or*two minute of audio recording done on VoiceThread, share what you learned about speakers and waves

**Option 2**: Investigate a fire-based audio visualizer- Watch a video about a fiery ‘Ruben’s Tube’ audio visualizer here: https://www.youtube.com/watch?v=BbPgy4sHYTw
- Homework:
*Either*through one page of writing, a conceptual map with 3-4 diagrams,*or*two minute of audio recording done on VoiceThread, explore*one*of the following:- How a Ruben’s tube works
- How a standing wave is produced
- How frequency and amplitude can change and be visualized in a Ruben’s tube

**Option 3**: Discover more about ocean waves, waves on a string, and light- Watch two videos on transverse waves, power, wave speed, frequency, and ocean waves here:
- Homework:
*Either*through one page of writing, a conceptual map with 3-4 diagrams,*or*two minute of audio recording done on VoiceThread, create a mini ad campaign informing the public about tranverse waves in and out of water

Draft two headlines expressing what we did today: one funny (but not vulgar) and one serious and captivating of what you learned in class today__PART 2__:

__Distribute__

__Web Resources__

- “Trig Mini Golf”: http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID2=AB.MATH.JR.SHAP.TRIG&lesson=html/object_interactives/trigonometry/use_it.html
- “Exploring Trigonometry (Video Interactive)”: http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID2=AB.MATH.JR.SHAP.TRIG&lesson=html/video_interactives/trigonometry/trigonometrySmall.html
- “Trig Tour”: https://phet.colorado.edu/sims/html/trig-tour/latest/trig-tour_en.html
- Interactive Unit Circle: https://www.mathsisfun.com/algebra/trig-interactive-unit-circle.html
- Transformations of Trigonometric Functions: https://illuminations.nctm.org/Activity.aspx?id=3589
- “Trig Spinner”: http://www.kwarp.com/portfolio/trigspinner.html
- Compare and contrast transverse, longitudinal, and periodic transverse waves: http://www.physics.nyu.edu/~ts2/Animation/Trans_Long_Periodic_Waves.html
- “Trigonometry: Easy to Understand 3D Animation”: https://www.youtube.com/watch?v=ovLbCvq7FNA
- “Ruben’s Tube Theory”: https://www.youtube.com/watch?v=BbPgy4sHYTw
- Sound Wave Interference: http://ccl.northwestern.edu/netlogo/models/Speakers
- PhET: “Waves on a String”: https://phet.colorado.edu/sims/html/wave-on-a-string/latest/wave-on-a-string_en.html
- “iPhone 4 Inside A Guitar Oscillation”: https://www.youtube.com/watch?v=INqfM1kdfUc
- The Superposition Principle: Interference Patterns: http://www.physics.nyu.edu/~ts2/Animation/Superposition.html

__Formative Assessments__

- Informal checks by the teacher as students engage in ‘Carousel’
- “Carousel Homework”

__Summative Assessments__

- K-W-L (‘
**K**now-**W**ant to Know-**L**earned’)