Cyclical Functions
Learning Module
- Creator(s):
The idea of new media has really opened the door to a unique type of literacy learning. Cope and Kalantzis (2015) have written that the current trend is a move away from media consumers and toward media producers. In order for our students to meet this challenge, there needs to be a cultural shift in our classrooms to adapt our practices to keep up with the changes. Students should move from passive meaning making and become active producers of meaning in collaboration with their peers.
In the video below, the concept of contemporary literacies is discussed.
Cope, B. & Kalantzis, M. (2020, February 23). [College of Education]. https://www.youtube.com/watch?v=VcfXSxchMnc&feature=emb_logo. Retrieved from youtube.com.
Contemporary literacies pedagogy has an emphasis on extending material to a create new meaning. We see this in the four quadrant picture discussed in the video and posted below for reference.
I find this especially true for the "applying creatively" and "experiencing the new" components, which can include media outside of the texbook. With contemporary literacies pedagogy, it is still important to read the text or make meaning of other forms of information, and the traditional literacy methods still have their place, however, there is a shift towards making new meaning and connecting concepts through a more diverse experience.
As a result, this work was created to include modules that focus on new meaning making in a variety of ways. Each of the four quadrants above are incorporated at some point in the updates. Each update module is created to be done in one 50 minute class period, with the end peer reviewed project as a summative assessment. This class is a third year high school math class and students will have already taken Algebra 1 and Geometry. This unit occurs in the second semester of the course and relies on the following pre-requisite material that students have already covered in prior units in this course:
In this module on Trigonometry, you will begin the unit with the exploration and creation of a new curve that will include several parameters we have studied within the family of functions. You already know that the "a" value of a function can reflect the function over the x-axis, as well as stretch or compress a graph. You have learned in a prior unit that the "h" parameter will shift a grpah horizontally and the "k" parameter will shift a graph vertically. For example in the equation y = 4 (x-3)2 +1, the graph has been stretched by a factor of 4, shifted to the right 3 units and shifted up 1 unit. These same parameters can be seen in trigonometric functions, however, you will also work to create a new parameter for the trigonometric functions. You will look for, and make use of, structure as you build connections between the sine and cosine graph and the unit circle.
As for standards, the class focus on the following from the Common Core State Standards listed below, which includes content standards and the math practice standards. These standards are what the state of Illinois uses to measure student growth.
F-TF.5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
F-IF.7e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
F-IF.9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
F-BF.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.
CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4 Model with mathematics.
CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6 Attend to precision.
CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.
The 6 updates listed below should be done in sequential order and can be a part of any junior level high school course. Each update would correlate with one 50 minute class period. There will be an assessment before the final project that will assess the learning process. The final project will be to create a new application that can be modeled by a cyclical function. The project will be peer reviewed and you will have the opportunity to revise the project based on feedback from your peers.
The sequence of information, and many of the examples and discussion points are taken from the College Preparatory Mathematics curriculum (www.cpm.org) that is used in my current school. I chose this topic because I am looking for a way to incorporate new media literacies and peer reviews into the current work in order to make this chapter more meaningful and useful for my students.
Content Goal: Students will conduct an experiment with a pendulum that will result in a sine curve.
Language Goal: Students will predict ways to change the shape of the curve.
An anticipation guide is similar to a pre-course survey. You will be given a list of statements that you will either mark True or False. From your anticipation guide, you may have an idea regarding our next area of study, and you will be able to preview some of the material that you may need to focus on. As an individual, complete the following anticipation guide to infomally assess your prior knowledge. We will complete the right hand side of the anticipation guide at the end of the module. Your teacher will give you feedback throughout the unit and you can make adjustments to your anticipation guide at any point.
Open the PDF to access the anticipation guide.
After the Anticipation Guide is complete, your group will complete the experiment below.
COMMENT: Make a post indicating your results and predictions for how to make a different shape with the graph. Comment on at least two of your peers' results with any similarity and differences compared to your response.
UPDATE: Find another example of something in real life that has the same shape as what you saw in class. Post a picture of this example and make connections between the shape and the context of the situation.
Lesson Objective: Students will conduct an experiment with a pendulum that will result in a sine curve.
CCS Standard(s): F-TF.5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
Mathematical Practices:
CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP4 Model with mathematics.
Lesson Overview: Update 1 can be organized as a whole class activity. This will reduce the amount of materials and prep time, and allow groups to observe the intended affects of the pendulum swing. Prepare bags ahead of time and include 3 or 4 so that you can run the experiment several times.
Before beginning the activity, have students make a guess as to what the dripping blood will look like. Keep in mind that students will not have formal vocabulary at this point and are not required to use terms like amplitude and midline.
After the experiment is complete, allow students time to make conjectures on how to change the shape of the curve. Details on the preparation of materials is below. Save several graphs to dry and hang up later so that you can add vocabulary to the picture.
Answer key: The following are all acceptable responses for the comment section:
Once students have submitted the update component, read through them and provide feedback. Group similar ideas together and have a class discussion on whether or not the findings model the same behavior.
Content Goal: Students will conduct new experiments with a pendulum that will result in a sine curve that has been stretched, compressed, reflected, and/or shifted horizontally/vertically.
Language Goal: Students will predict ways to change the shape of the curve.
Yesterday, you completed an experiment with Nurse Nina's blood path and analyzed the shape that was created. Today, we will extend your thinking to what other graphs you can make. As you work with your team, think about how create new graphs. Questions to prompt your learning are:
How long do you want each cycle? How tall do you want it to be? Where do you want it to be on your paper? Where do you want it to start?
Predict how you could conduct an experiment to get exactly the curve you have described. How fast should you move the paper? Where would you start it? How high would you start your pendulum?
Complete the experiment again to test your hypothesis.
Comment: On the community page, note your observations and the equations you have created. Verify that at least two of your peers' equations would also work within the context.
Update: Do any of the functions you are already familiar with model this situation? Why or why not?
Lesson Objective: Students will predict and test ways to change the shape of the curve.
CCS Standard(s): F-TF.5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
Mathematical Practices:
CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them. CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4 Model with mathematics.
For the second day, allow students time to create new graphs based on length and speed of pendulum swing, rate of paper moving, distance between maximum and minimum heights, starting position of the paper, starting position of the bag, rate of liquid leaving the bag, or any other ideas that were posted from Update 1.
As they discuss this, expect to hear statements such as, “If we want to change… we should…” and “If we changed… the curve should…”. Encourage students to be meaning makers and try new things. There are no right or wrong answers here, the point is to let them create and discover!
Content Goal: Students will use experimental data generated from measuring the heights of right triangles to create a sine graph.
Language Goal: Discuss how the new graph may be shifted using a previously learned parameter.
Today you will use what you know about right triangle trignometry to create a new function. Read the prompt below and then complete the task on Desmos with your group.
Your task is to write a function that describes how far above or below the platform a rider is when The Screamer breaks down. Use the Desmos tool here https://www.desmos.com/calculator/kg2pbzkewx to gather data. Once you have gathered the data, plot it on an x-y coordinate plane and note any observations.
Comment: After completing the activity, post the answers to the following questions on the community board:
1. Was it necessary to measure all heights? Why or why not
2. From one specific height, can you make any predictions about any riders who are at other locations on the Ferris wheel?
3. What patterns did you observe?
Lesson Objective: Students will use experimental data generated from measuring the heights of right triangles to create a sine graph.
CCS Standard(s): F-TF.5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
Mathematical Practices:
CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP4 Model with mathematics.
Lesson Overview: Today's update gives students an opportunity to make connections between the unit circle and the angle or rotation that is created by generating and anaylying data from a real life application. They will use technology to model the data and look for and make use of patterns. They will graph the angles as the independent variable and the heights as the dependent variable and use their prior knowledge of right-triangle trigonometry to name this function h(θ)=sinθ. They will begin to make connections to the swinging back and forth of the blood ba, the sine curve and the cyclical nature of the unit circle.
The Screamer is a special Ferris wheel that has its horizontal axis at ground level, so any given seat is above ground level half of the time and below ground level the other half of the time. Students will measure the height of one seat on the wheel as it is being rotated. They will graph the angles as the independent variable and the heights as the dependent variable and use their prior knowledge of right-triangle trigonometry to name this function h(θ)=sinθ.
Students will continue to make label multiple heights in a unit circle and will graph these heights onto a set of axes to make a sine graph. The image below is what the graph should look like:
Teachers, note the responses and correct any misconcpetions posted on the community board. You may wish to have one-on-one conferences with students to solidify their understanding and/or clear up any confusion that exists.
Content Goal: Students will develop an understanding of reference angles
Language Goal: Students will explore the connections between the sine graph and the unit circle.
Watch the video below. Your work over the prior updates has shown that relationships between the sine graph and the unit circle exists. As you watch the video, pay attention to the angles of rotation, the height of the sine curve and the position of the point on the unit circle.
Comment: What do you notice? Post a comment on the community board with at least two observations.
Etkin, A. (2020, February 23). [Arkady Etkin]. https://www.youtube.com/watch?v=Q55T6LeTvsA. Retrieved from youtube.org
Now, watch this video on reference angles. After the video, find the reference angle for the following degree measures and submit to me via google classroom as a formative assessment.
McLogan, B. (2020, February 23). [Brian McLogan]. https://www.youtube.com/watch?v=Flx0m5CIrpE. Retrieved from youtube.org
Lesson Objective: Students will develop an understanding of reference angles and will explore the connections between the sine graph and the unit circle.
CCS Standard(s): F-IF.7e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
Mathematical Practices:
CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.
Lesson Overview: Have students watch the sine/unit circle video to solidify their learning from the prior updates. At this point in the learning, it is very important that students make the connection that the unit circle and the sine graph are both ways to represent the funciton y = sin(x). Student will also learn what a reference angle is and how to find the value of a reference angle from 0 degrees to 360 degrees. The correlation of the four reference angles (one in each quadrant) should be made in relation to the heights of the Screamer riders and the y coordinate of the unit circle.
When students submit their formative assessment on google classroom, use the following key:
300 degrees = 60 degree reference angle
-210 degrees = 30 degree reference angle
50 degrees = 50 degree reference angle
270 degrees = cannot solve since this is an axis and you cannot create a triangle with a height perpendicular to the x-axis.
Content Goal: Students will determine the placement of the parameter b in the general equation for sine and cosine.
Language Goal: Students will identify what effect b has on the graph
We already know that the three parameters a, h, and k move our graphs in different ways from our studies in the earlier chapters in this course. We have one final piece to analyze when it comes to transforming the sine graph, and this is our missing 4th parameter, b. Use the Desmos graphing tool found here https://www.desmos.com/calculator to investigate what this fourth parameter does. In other words, what affect does it have on the sine graph and how can we create an equation to model this.
Comment: After you have experimented with this new movement, post to the community and offer feedback to others' hypotheses.
Lesson Objective: Students will determine the placement of the parameter b in the general equation for sine and cosine.
CCS Standard(s): F-IF.7e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
F-IF.9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
F-BF.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.
F-TF.5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
Mathematical Practices:
CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.
Lesson Overview: Students will investigate the fourth parameter of trigonometric functions. This will be an investigation using Desmos and will afford students the opportunity to make new meaning of the movement of the graph and the horizontal compression that cyclical functions have. The b value causes the period of the sine functions to expand or condense. In the graph y=sinx, the period is 360 degrees, meaning one full since curve can be found from 0 to 360 degrees. When the b parameter is included, this tell us how many full sine curves are in between 0 to 360. For example, y = sin2x, there would be two full curves from 0 to 360.
Content Goal: Students will combine all ideas to create a project reflecting a cyclical model as learned throughout this module.
Language Goal: How can you use what you have learned and apply it?
For your final project you are to create the following. Be sure to use multimedia elements in your project.
Explain why are the trig function (sine, cosine, and tangent) considered periodic, also known as cyclical? Create a situation that has a periodic structure and model it with a trigonometric function. Be sure your explanation is clear and complete. Include at least 4 multimedia elements and multiple representations such as equations, tables, and graphs.
You will receive feedback from your peers based on the PDF of the rubric below. Once you get feedback from your peers, you may revise your work and resubmit before the teacher scores your work.
Lesson Objective: Students will combine all ideas to create a project reflecting a cyclical model as learned throughout this module.
STANDARDS: all listed previously
MATH PRACTICE STANDARD: model with mathematics
This final update pulls together all the things learned in the module into a project. Student create a situation and model it using trigonometry. Final project will be submitted with multimedia elements and will be peer reviewd with a rubric. Once students have received peer feedback, they may revise before submitting for a final summative grade on the unit.
Cope, B., & Kalantzis, M. (2015). The things you do to know: An introduction to the pedagogy of multiliteracies. Palgrave Macmillan. https://doi-org.proxy2.library.illinois.edu/10.1057/9781137539724
Cope, B. & Kalantzis, M. (2020, February 23). [College of Education]. https://www.youtube.com/watch?v=VcfXSxchMnc&feature=emb_logo. Retrieved from youtube.com.
Cope, B. & Kalantzis, M. (2020, February 23). [College of Education]. https://www.youtube.com/watch?v=VcfXSxchMnc&feature=emb_logo. Retrieved from youtube.com. [Image, screenshot].
Core Connections Algebra 2. (2013). Retrieved from www.ebooks.cpm.org
Earlham College. (n.d). Sine Curve [digital image]. Retrieved from https://legacy.earlham.edu/~tobeyfo/musictechnology/2_SineWaveMath_edit.html
Etkin, A. (2020, February 23). [Arkady Etkin]. https://www.youtube.com/watch?v=Q55T6LeTvsA. Retrieved from youtube.org
Fox, S. (2019). Missing. [digital image]. Retrieved from http://shaunfox.com/work/something-missing.html
McLogan, B. (2020, February 23). [Brian McLogan]. https://www.youtube.com/watch?v=Flx0m5CIrpE. Retrieved from youtube.org