Teachers in Industry

University of Arizona College of Education
Teachers in Industry / 1
Stage 1 – Desired Results
Unit Summary:
Algebra 2 – Unit Circle and Trig Functions- Understanding Periodic Relationships
Unit Time Frame: 6 days with block scheduling
Relation to Workplace: The power cycle of a nuclear reactor core for a 48 month period, and the human impact it has through the cycle of events required between startup and shutdown.
Established Goals
F-IF.B.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. («)
F-IF.C.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. («)
e.  Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
F-BF.B.3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
F-TF.A.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the arc.
F-TF.A.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
F-TF.B.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
Math Practices:
MP.1 Make sense of problems and persevere in solving them.
MP.2 Reason abstractly and quantitatively.
MP.4 Model with mathematics.
MP.7 Look for and make use of structure.
MP.8 Look for and express regularity in repeated reasoning.
**Consider including electricity phases from science standards. / Transfer
·  Understand how trigonometric functions can represent and explain cyclical patterns that occur around and within us.
·  . Model cyclical patterns with trigonometric functions and their graphs to solve application problems.
Meaning
Understandings
·  The relationships in periodic systems are patterns that can be modelled with a trigonometric function.
·  The components of the unit circle relate to periodic functions and their key points.
·  Periodic functions can be identified by their frequency, and periodicity.
·  Periodic functions can be used to help predict future events and what can be expected to occur before/or afterwards. / Essential Questions
How can understanding cyclical/periodic systems be useful for decision making?
·  What relationships can be found between cyclical/periodic systems?
·  What can be interpreted from key features of a graphed cyclical/periodic system?
Acquisition
Students will know …
·  The intervals of vertical height around the unit circle can be used to create a sine function.
·  The intervals of horizontal width around the unit circle can be used to create a cosine function.
·  Key points of a periodic graph are the intercepts, localized max and min, the period between each cycle, and the frequency of cycles within a unit.
·  Vocabulary: sine, cosine, tangent, radian, unit circle, period, frequency, amplitude, midline / Students will be skilled at
·  Analyze and interpret trigonometric functions regarding key information.
·  Use the unit circle to create a trigonometric function.
·  Work with others to creatively develop a model of a cycle found in their life using a trigonometric function.
·  Make predictions about periodic functions based on their graphical models.
·  Solve non-routine problems with innovation and accuracy.
·  Ask reflective questions that guide and refine their work.
·  Communicate ideas effectively in a variety of situations.

***All highlights in green are updates/changes made to stage 1 according to previous feedback provided.

Stage 2 - Assessments
Performance Task (in GRASPS format)
To present a model using trigonometric functions to demonstrate two cyclical patterns found in an industry and analyze the relationships between the initial pattern modeled, and the second pattern modelled for the adjustments made to represent a predicted event that could impact the industry in the future.
Students are a team of analysts and engineers (R) from a small startup consulting company given an opportunity to land a big deal with a major industry client. The team will determine and model two cyclical/periodic patterns related to the industry(G). These patterns will be modeled from data collected through research of the selected industry. The data will need to be graphed and modeled with a trigonometric function.
Examples of Periodic Patterns Found in Industry:
Traffic Patterns on Freeways, Number of Accidents, Airflight Traffic (Amount of Activity vs. Time of Day)
Mail Delivery, Taxi Rides, Rental Movie Times, Criminal Activity for Youth (Amount of Activity vs. Month or Time of Day)
Technology Purchases, Car Purchases, School Supply Purchases (Amount of $ Spent vs Month)
Employees Hired (Number of New Hires or Total Employees vs. Month)
High Temperatures, Average Temperatures, Amount of Rainfall, Number of Storms ( ___ vs. Month)
Examples of an Event that Changes the Industry Pattern:
Self Driving Cars, Public Transportation Changes, Traffic Cameras, Change in Speed Limits, New Trucking Route, Larger Aircrafts, New Airport
New Youth Center, Year Round School, New After School Programs
New Technology Release Schedule, New Method for Purchasing, New Sale Strategy
Growth or Reconstruction of Company
Global Warming, Change in Air Patterns, Deforestation
Impacts of Natural Disaster (Flood, Fire, Earthquake, Tsunami, Volcanic Activity, Hurricane, Tornado, etc.)
Expectations for Analysis:
·  Calculations must explain how the period, and frequency of how both Trigonometric models were determined and demonstrate with evidence that their model is a “good” approximation of the data.
·  Analysis must include a description of the relationships found between key points of the graphical models and interpret how these relate to the industry overall.
·  The team must forecast the occurrence of an event that will impact the future of the industry and apply the appropriate transformations to adjust their model of the function and graph in order to depict the impact of the event on the industry.
·  Additional analysis should explain how these predicted changes and transformations would change the relationships between the models and their key points. Also considering the perspective of the industry a recommendation should be prepare for how the industry should respond to the potential event.
Industry managers (A) who are responsible for future planning in the industry will be considering the submitted models, predictions, and recommendations in order to determine if they should make adjustments to prepare for the predicted event that has been brought to their attention. The consultant team needs to convince the managers to follow their recommendations and secure future work with the industry. (S)
A final presentation (P) will be required to explain your models, predictions, and recommendations regarding the impact of the industry event. The presentation must clearly demonstrate the expectations for analysis already described and communicate effectively the problem, relationships, and changes that are represented by the models. (S)
Other Evidence: (quizzes, tests, prompts, work samples, labs, etc.)
Prompt: Consider a cyclical pattern related to your life, sketch a graph to model the pattern and calculate a trigonometric function that would produce the same graphical model. Explain the key points of the graph in relation to the context of the pattern modeled. (Key points: intercepts, max, min, domain, range, period, frequency)
Activities:
Biorhythms Graphs (Extension: Circadian Rhythms)
Spaghetti Sine and Cosine Waves Activity
Desmos Polygraph Periodic Functions or Trigonometry Function Family Art Project
Extensions: Circadian Cycles, Mathalicious – Sound of Silence
Quiz 1: Unit Circle and Radians
Quiz 2: Transformations of Periodic Functions
Project Related:
Project Rubric
Peer Feedback Form – Presentations
Individual Self-Assessment and Reflection
Group Action Plans
Ongoing:
Formal Checks for understanding (exit tickets, project pacing guide, status reports, reflections)
Informal Checks for understanding (questioning, observations, and discussions)
Student Self-Assessment and Reflection:
Individual Reflection:
Students will reflect individually through exit slips after class activities or assignments (ie. What about this topic do you understand now that you didn’t before? What about this topic would you like to see discussed in class again?)
Students will also individually reflect on their contribution to the project activity and provide a summary of what they learned from the unit, and the project, what grade they would give themselves based on their understanding, participation, and overall excellence.
Students will reflect on their graded assignments afterwards with a goal written down before entering into the next unit.
Peer-Reflection:
Students will have opportunities to provide informal peer-reflections after sharing ideas in class, and after collaborating on activities.
Students will also provide peer feedback to groups for each project presentation using a rubric.
Students in project groups will reflect daily at the end of class to identify their “status” for their work and submit this to their teacher.
Students in their project groups will collaborate after a project has been completed and they have received and reviewed the peer and teacher feedback, and submit a summary of what they have learned from the feedback they received and a list of actions that they would set as goals to improve their project for next time.
Stage 3 – Learning Plan
Day 1: Introduction to Period Functions
(H)Begin by showing students the modeling video of Reuben Margolin: On Kinetic Art to pose the question, “What in life has highs and lows, is always changing but is always the same?”
https://www.youtube.com/watch?v=D2HF-1xjpP85:40 to 8:14, 9:00, 11:00, 17:46
(W, R) Some discussion will be prompted after the excerpts of the video, using whole class discussion and think pair share methods. “What terms did you notice used that were mathy?” “What was the significance of the artwork that was displayed?”
(E, R, O)Students will individually consider the following prompt, develop a response independently, then they will share their graph with their small group, and discuss how their model demonstrates a cyclical pattern. Members of the group will need to ask clarifying questions, and provide positive constructive feedback for each other.
Prompt: What cyclical pattern do you know occurs around you? Sketch a graph of something measurable that would demonstrate the pattern along a time line.
(E) Concept Developing Activity: (See Appendix: Day 1 Biorhythm Activity)
Students will work with a partner to each complete their own Biorhythm graph and interpret the differences in the patterns in the graph and compare the differences found in the functions.
(R, E) Students will reflect on their activity work compared to the graphical model created for them at http://www.intmath.com/trigonometric-graphs/biorhythm-graphs.php

(T) Extension: For early finishers is to answer application questions about patterns related to sleep: http://www.biology.arizona.edu/biomath/tutorials/Trigonometric/Applications/CR.html
(E, R, E) Exit Slip: Individual students will demonstrate their understanding of modeling cyclical patterns graphically and using functions in the Ferris Wheel Activity (See Appendix. Day 1: Exit Slip)
Day 2: Introduction to the Unit Circle
(W, T, O) Pre-Assessment: Students will be assessed (See Appendix. Day 2: Introduction to the Unit Circle Pre-Assessment) on their understanding of right triangles and trigonometry from geometry. Based on the students understanding additional instruction may be included to refresh students on trigonometry prior to beginning the activity.
(R) Instruction: Teacher will use direct instruction (~15 minutes) to connect the pre-assessment material of the geometric concepts of trigonometry in right triangles to the unit circle. Using questioning methods and small group discussions to ensuring that students recognize the pattern between the coordinate points in each quadrant of the plotted unit circle and the relationship to the length and height of each triangle within the circle to the sine and cosine of the central angles.
(H, E) Interactive Presentation: Teacher will introduce the concept of radians. Using a PearDeck where students will read, and demonstrate their understanding of the explanations found on the following sites by answering questions and creating their own explanation during the presentation. https://mathwithbaddrawings.com/2013/05/02/degrees-vs-radians/
http://betterexplained.com/articles/intuitive-guide-to-angles-degrees-and-radians/
http://www.mathopenref.com/radians.html
Teacher will use PearDeck to check for understanding and refine the students’ ideas before starting the next activity. (see www.peardeck.com)
(H, E, T, O) Concept Developing Activity: Students will work in small groups to complete the spaghetti sine wave project using their understanding of the unit circle to build the sine and cosine function with pieces of spaghetti based on the height or length of each triangle created in the unit circle. (See Appendix Day 2: Spaghetti Sine Wave Activity)

(R, E) Students will complete reflection questions to demonstrate their understanding. (See Appendix. Day 2: Spaghetti Sine Wave Activity Questions)
(T) Extension: To further the discussion about the shifts and transformations of the functions for early finishers use the practice pages found at the end of the activity. (See Evernote for full file. http://www.evernote.com/l/AAi9yoqVhzJIt7L1sBqKylgIKKKsuNuTk1M/ ) or the matching activity found in lesson from http://map.mathshell.org/lessons.php?collection=8&unit=9255.
(E) Exit Slip: Students will be asked to reflect on their learning by answering the following questions.
- What about this topic do you understand now that you didn’t before?
- What about this topic would you like to see discussed in class again?
Day 3: Transformations of Trig Functions
(R, E) Quiz: Assess students on their understanding of the Unit Circle, Coordinates, and Radian (Short skill based quiz <10 questions) Students will be given opportunity to retest on Day 4 and Day 5 until they reach mastery.
(W, E, R) Reteach: Teacher will briefly reteach and refine students understanding from spaghetti functions and unit circle based on feedback from the quiz, and exit slips given the day before. Direct instruction should include creating a set of notes to organize the forms of the transformations that take place using trigonometric functions. Key vocabulary such as period, amplitude, phase shift, midline, and frequency should also be formally defined.