Connecting Graphs and Equations of Sinusoidal Functions
Connecting Graphs and Equations of Sinusoidal Functions ajlill Sun, 07/18/2010 - 18:12Rationale
The universe if full of phenomenon that are cyclical or follow a repeating pattern. Many of these can be modeled using sinusoidal functions.
Expectations
2.1 describe key properties of periodic functions arising from real-world applications given a numeric or graphical representation
2.2 predict, by extrapolating, the future behavior of a relationship modeled using a numeric or graphical representation of a periodic function
2.3 make connections between the sine ratio and the sine function and between the cosine ratio and the cosine function by graphing the relationship between angles from 0º to 360º and the corresponding sine ratios or cosine ratios, with or without technology, defining this relationship as the function f(x) =sinx or f(x) =cosx, and explaining why the relationship is a function
2.4 sketch the graphs of f(x) =sinx and f(x) =cosx for angle measures expressed in degrees, and determine and describe their key properties
2.5 determine, through investigation using technology, the roles of the parameters a, k, d, and c in functions of the form y =af (k(x – d)) + c, where f(x) =sinx or f(x) =cosx with angles expressed in degrees, and describe these roles in terms of transformations on the graphs of f(x) =sinx and f(x) =cosx
2.6 determine the amplitude, period, phase shift, domain, and range of sinusoidal functions whose equations are given in the form f(x) = asin(k(x – d)) + c or f(x) = acos(k(x – d)) + c
2.7 sketch graphs of y = af (k(x – d)) + c by applying one or more transformations to the graphs of f(x) =sinx and f(x) =cosx, and state the domain and range of the transformed functions
2.8 represent a sinusoidal function with an equation, given its graph or its properties Sample problem: A sinusoidal function has an amplitude of 2 units, a period of 180º, and a maximum at (0, 3). Represent the function with an equation in two different ways.
Settings
Students
The students are expected to have completed Section 1, Representing Functions and Section 2 Solving problems using quadratic of Characteristics of Functions, and so be familiar with the form y=af(k(x-d)) + c, and interpreting the meaning of and applying the transformations represented by a, k, d and c with respect to quadratic functions.
Teacher
The teacher should be familiar with the graphing and data manipulation software available in the school.
Classroom
Ensure LiveMath Viewer is available on the computers.
Assessment
Summative results will be from the Trig-Transform project and the Sunspots paper
Task Overview
- Spaghetti Triganometry Students create graphs of sine and cosine functions using spaghtti and string. - 1 day
- Play with transformations - vocabulary - graph to equation, equation to graph - amplitude, period and phase-shift exercises from Interactive Maths amplitude, period, phase shift activities. - 2 days
- Group-Graph Challenge - go from graphs to equations to properties and back - 2 day
- Trig-Transformers presentation - 3 days
- Sunspots - find amplitude, period of sunspot data - predict next cycle - 4 days
Spaghetti Trigonometry
Spaghetti Trigonometry ajlill Sun, 07/18/2010 - 18:56Specific Expectations
2.3 make connections between the sine ratio and the sine function and between the cosine ratio and the cosine function by graphing the relationship between angles from 0º to 360º and the corresponding sine ratios or cosine ratios, defining this relationship as the function f(x) =sinx or f(x) =cosx, and explaining why the relationship is a function
2.4 sketch the graphs of f(x) =sinx and f(x) =cosx for angle measures expressed in degrees, and determine and describe their key properties
Materials
Spaghetti, yarn, paper, glue.
Important Terminology
Sine, Cosine, Unit Circle, cycle, domain, range, intercepts, amplitude, period, maximum and minimum values, increasing/decreasing intervals
Background Knowledge
An understanding of the definitions of the sine and cosine ratio.
Lesson Sequence
Follow lesson 1 of Spaghetti Trig, using angles in degrees instead of radians. Split the students into groups of 2 or 3, have half the students do sine and half cosine.
Key Questions
What is the relationship between the graph and the sine ratio? Is it a function? Why is it called a "unit circle", what are the cycle, domain, range, intercepts, amplitude, period, maximum and minimum values, increasing/decreasing intervals
Assessment
Formative: Have the students answer the key questions in their math journals
Enrichment
Those who are finishing early can add more angles to their project.
Remediation
Extra time can be given to students who need it. Since it's group work, you can match students with difficulties with students that can support them.
Implications for Future Lessons
InterMath Exploration
InterMath Exploration ajlill Mon, 07/19/2010 - 18:16Specific Expectations
2.5 determine, through investigation using technology, the roles of the parameters a, k, d, and c in functions of the form y =af (k(x – d)) + c, where f(x) =sinx or f(x) =cosx with angles expressed in degrees, and describe these roles in terms of transformations on the graphs of f(x) =sinx and f(x) =cosx
2.4 sketch the graphs of f(x) =sinx and f(x) =cosx for angle measures expressed in degrees, and determine and describe their key properties
Materials
Internet, graph paper or a printer, radian to degrees cheat-sheet
Important Terminology
amplitude, period, frequency, phase shift, phase angle,
Background Knowledge
Hook
Lesson Sequence
- Review the actions of a, k, c, and d on a quadratic function.
- Review the properties domain, range, intercepts, maximum and minimum values, increasing/decreasing intervals).
- Have the students predict what effect the parameters a, k, c, and d have on the graph y=a sin(k(x+d)) +c and it's properties and record it in their journals
- Go over the relationship between radians and degrees
- Have the students complete sections 1, 2, and 3 from Graphs of the Trigonometric Functions In additions, have them list cycle, domain, range, intercepts, amplitude, period, maximum and minimum values, increasing/decreasing intervals for their graphs.
- Have the students compare what they've learned with what they predicted.
Key Questions
Leading questions given by you to the students and expected questions from the students
Assessment
Formative - collect the graphs they produce
Enrichment
Those who finish early can work on the remaining sections of the webpage.
Remediation
THere is also an excellent app at GeoGebra that lets you play with all the trasforming variables for any trig function.
Implications for Future Lessons
Group-Graph Challenge
Group-Graph Challenge ajlill Mon, 07/19/2010 - 18:19Specific Expectations
2.4 sketch the graphs of f(x) =sinx and f(x) =cosx for angle measures expressed in degrees, and determine and describe their key properties
2.6 determine the amplitude, period, phase shift, domain, and range of sinusoidal functions whose equations are given in the form f(x) = asin(k(x – d)) + c or f(x) = acos(k(x – d)) + c
2.7 sketch graphs of y = af (k(x – d)) + c by applying one or more transformations to the graphs of f(x) =sinx and f(x) =cosx, and state the domain and range of the transformed functions
2.8 represent a sinusoidal function with an equation, given its graph or its properties Sample problem: A sinusoidal function has an amplitude of 2 units, a period of 180º, and a maximum at (0, 3). Represent the function with an equation in two different ways.
Materials
Pencil, graph paper, cards with starter problems
Important Terminology
domain, range, amplitude, period, phase shift.
Background Knowledge
Hook
Lesson Sequence
Break the class up into groups of 3. Distribute problem cards. Each student in the group picks a card. Each card will have either a graph, a function, or a list of properites. Each student takes whichever card they are given, and creates the missing two cards. They then pass the cards to their group mates, who repeat the process. They then compare their answers. Repeat once more, then have the students make up their own problem cards. Return the original card. This leaves 3 pair of cards showing the same function. Have the students from groups of 6, and assemble 3 decks of cards, each deck containing 1 pair from each function, but not identical, i.e. one pair can consist of a graph card and an equation card. Keep 1 deck, pass one to the group to your right, and one to the group on your left. Play go fish with the cards.
Key Questions
Leading questions given by you to the students and expected questions from the students
Assessment
Students can do a self assessment or mark each others work.
Enrichment
Remediation
Implications for Future Lessons
Trig-Transformers Presentation
Trig-Transformers Presentation ajlill Mon, 07/19/2010 - 18:20Specific Expectations
2.5 determine, through investigation using technology, the roles of the parameters a, k, d, and c in functions of the form y =af (k(x – d)) + c, where f(x) =sinx or f(x) =cosx with angles expressed in degrees, and describe these roles in terms of transformations on the graphs of f(x) =sinx and f(x) =cosx
Materials
art supplies, computer access
Important Terminology
Background Knowledge
Hook
Lesson Sequence
Have the students from groups or 3 or four, and task them with producing a poster/presnetation/website/whatever that demonstrates the effects of a, k, c and d on the graphs of f(x) = a sin( k (x + d ) ) + c, and on it's domain, range, amplitude, period, cycle, phase shift.
Key Questions
Leading questions given by you to the students and expected questions from the students
Assessment
Their product is a summative assessment of their understanding of the properties of snusoidal functions.
Enrichment
Remediation
Implications for Future Lessons
Sunspots
Sunspots ajlill Sun, 07/18/2010 - 23:27Specific Expectations
2.1 describe key properties of periodic functions arising from real-world applications, given a numeric or graphical representation
2.2 predict, by extrapolating, the future behavior of a relationship modeled using a numeric or graphical representation of a periodic function
Materials
Computer Lab
Important Terminology
amplitude, period, sunspot, solar cycle
Background Knowledge
None beyond previous lessons in unit.
Familiarity with software available to turn data to graphs - add a demonstration to the start of the computer portion of the lesson if necessary.
Hook
Lesson Sequence
1 - Explain how sunspots come in cycles and that they will be using real data to build a model of the sunspot or "solar cycle" and write a report on it. - 10 minutes
2 - Distribute handout or point them to the location on the class webpage where they can find it. Hand out an exemplar paper or an article from "Scientific American" or a similar magazine. Discuss the assignment - 10 min
3 - Move to computers and have them create their graphs - remainder of period.
4 - allow an additional 2 - 3 periods in computer lab to gather info and complete papers.
Key Questions
Leading questions given by you to the students and expected questions from the students
Assessment
The paper they write will be marked according the the attached rubric for their summative assessment
Enrichment
Is there a larger pattern visible in the sunspot data. How could you improve your model.
Remediation
Extra time can be given to those sudents that need it.
A Mathematical Model of Sunspot Activity Handout
A Mathematical Model of Sunspot Activity Handout ajlill Mon, 07/19/2010 - 20:24The purpose of this project is to produce a report on the sunspot cycle and how you developed a mathematical model of it. The report should be 2-3 pages long and include
- a graph of the data,
- your mathematical model and an explanation of the amplitude, period, phase shift and offset,
- a story of how you developed and tested it.
- A discussion on how well your model matches the data and what you could do to improve the model.
- A discussion of why or why not your model matches the current sunspot cycle.
- A prediction of the time and intensity of the next solar maximum.
- Why we should care about sunspots.
Include a list of references and pointers to further information on the subject.
Instructions
Download the data set of daily sunspot numbers since 1818 from the Solar Influences Data Analysis Center (SIDC) http://sidc.oma.be/ and graph it. You can do this step with a partner.
Find a sinusoidal function that best matches the resultant graph. The function should take a time as input and return the predicted number of sunspots.
Check your model's accuracy by comparing the expected results with the actual results from the previous solar cycle and explain any discrepancies. Do the same with the current cycle.
Perform any additional research needed to answer the questions.
Assessment
You will be marked using a checklist of the required contents, as well as the following rubric.
| Criteria | Level 1 | Level 2 | Level 3 | Level 4 |
|---|---|---|---|---|
| Use of planning skills (e.g., generating ideas, gathering information, focusing research, organizing information) | uses planning skills with limited effectiveness | uses planning skills with some effectiveness | uses planning skills with considerable effectiveness | uses planning skills with a high degree of effectiveness |
| Use of processing skills (e.g., drawing inferences, interpreting, analyzing, synthesizing, evaluating) | uses processing skills with limited effectiveness | uses processing skills with some effectiveness | uses processing skills with considerable effectiveness | uses processing skills with a high degree of effectiveness |
| Use of critical thinking processes (e.g., research, critical analysis, critical literacy, metacognition) | uses critical thinking processes with limited effectiveness | uses critical thinking processes with some effectiveness | uses critical thinking processes with considerable effectiveness | uses critical thinking processes with a high degree of effectiveness |
| Expression and organization of ideas and information (e.g., clear expression, logical organization) | expresses and organizes ideas and information with limited effectiveness | expresses and organizes ideas and information with some effectiveness | expresses and organizes ideas and information with considerable effectiveness | expresses and organizes ideas and information with a high degree of effectiveness |