L1+Emery,+Jordan

UNIVERSITY OF MAINE AT FARMINGTON COLLEGE OF EDUCATION, HEALTH AND REHABILITATION**
 * UNIVERSITY OF MAINE AT FARMINGTON
 * LESSON PLAN FORMAT**
 * __Teacher’s Name__****:** Ms. Emery** __Date of Lesson__:** Lesson 1 (Application)
 * __Grade Level__****: **10 ** __Topic__:** Functions

Student will understand that functions can be expressed in several ways. Student will know the definitions of “domain”, “range”, “ordered pair”, “function” and “vertical line test”. Student will be able to recognize the graph and sketch graphs of basic functions.
 * __Objectives__**

Maine Learning Results: Mathematics - D. Algebra Functions and Relations Grades 9-Diploma Students understand and interpret the characteristics of functions using graphs, tables, and algebraic techniques. a. Recognize the graphs and sketch graphs of the basic functions
 * __Maine Learning Results Alignment__**


 * Rationale:** Students will be exposed to elementary functions in the forms of equations and graphs. Students will become with familiar with both forms of functions and recognize how to model functions in various ways.

Students will use a variety of methods throughout the lesson to solidify their understandings with the material presented to them on functions. First, students will use a 3-minute review with a peer to compare charts and add/change any information on their graphic organizers. They will review the material that they have gathered on functions in this time frame and re-evaluate any of the material that they have included on their graphic organizer. After they work with each of the functions in the groups and during instruction, students will return to the "hook" and match the music clip description with the graph of the function. By doing this, they will reconfirm the material was presented to them in a different way. Students will be able to return to their graphic organizer to add/change details once the activity is complete. After they have completed several in-class formative assessments with peers, the teacher will provide students will feedback on organizers so that they will be able to produce a good e-folio. Finally, after students complete their lesson on recognizing graphs, they will complete a checklist of their understandings. There will be a space on their checklist of understandings for additional comments where they can write if they have any other questions.
 * __Assessment__**
 * Formative (Assessment for Learning)**

Students will be asked to create an electronic portfolio using Noteshare that demonstrates their understandings on functions. In this e-folio students will be asked to explain and provide examples of the terms: domain, range, ordered pair, function, and the vertical line test. Then, students will be asked to explain the shape of the following function and create a graph in Geogebra. The product is an e-folio. Students will be assessed using a checklist that determines whether students have appropriately addressed each of the components that is to be included in the e-folio.
 * Summative (Assessment of Learning)**

__Technology__: Students will be asked to create an e-folio in which they are to create artifacts (word documents, GeoGebra photos, music clips) that demonstrates their ability to identify and explain each of the elementary functions addressed in the lesson. They will also be asked to listen to music clips and be exposed to graphing software to help them throughout the lesson.
 * __Integration__**

__Music__**:** Music clips are going to be use to represent the graphs of each of the elementary functions and provide students with another tool for recognizing each of the elementary functions

__English:__ Students will be asked to compose a letter and summarize their learning in creating the e-folio of this lesson.

Students will use a 3-minute review with a peer to compare charts and add/change any information on their graphic organizers. Students will be paired by being handed an index card when they walk into class (I will have already pre-determined partners, but students will not know this). After being presented with the material, students will need to identify their function and find a student that has the matching function. This will be their partner in the 3-minute review exercise. When students are in this exercise, one student will take the role of "analyzer" and the other will take the role of "journalist". The analyzer will compare both charts and compare the similarities and differences in the two graphic organizers. The "journalist" with record the similarities and differences. Then, both the journalist and the analyzer will work together to determine which information should be changed or fixed on both graphic organizers.
 * __Groupings__**

· __Verbal__: I will be directing students on how to recognize graphs, what the graphs look like, and how to complete tasks all in a verbal manner. · __Logical:__ Students are in a Logical/Mathematical setting in which students need to use mathematics skills to compute answers and process information. · __Visual__**:** Graphs of each of the functions are going to be on the screen (produced with GeoGebra) so that students will have a visual demonstration of what each looks like. · __Bodily/Kinesthetic__**:** Students will act out "domain", "range", "ordered pair", and "vertical line test". · __Musical__**:** Students will listen to music clips that represent the shape of functions. · __Intrapersonal:__ Students work individually to fill out graphic organizer and process the information. · __Interpersonal:__ Students will work in cooperative learning to clear up misconceptions with understanding functions as well as when they revise their thinking about functions. I will review student’s IEP, 504 or ELLIDEP and make appropriate modifications and accommodations.
 * __Differentiated Instruction__**
 * Strategies**
 * Modifications/Accommodations**

If a student is absent during this lesson, I will ask them to come meet with up when they return to school. Since the musical aspect of this lesson really solidifies the concept, they will asked to listen to the audio clips and fill out the graphic organizer just as their peers did. I will ask the students to gather notes on the shapes of the functions from a peer, and gather notes that were taken in class from a responsible classmate. The student will be given all necessary handouts that were provided in class for this lesson, and they can consult notes, peers and myself to complete the necessary assignments. A wiki page will be accessible so that they will be able to be current with their lessons.

Students are going to produce an e-folio at the end of this lesson. The e-folio is a Type II technology because students are given the task of creating evidence of their understandings for a variety of functions. Students will need to create meaningful artifacts that clearly demonstrate their understanding and ability to recognize the material. Students will use GeoGebra, word processing pages and Inspiration to create artifacts that demonstrate their new abilities in function recognition. Students will be asked to create their own examples, so the information becomes embedded in them and their product is reflected in a Type II way. Students will use their personal preferences to create these artifacts and organize them electronically in a way that best makes sense to them. Since it is a Type II technology, students are at an advantage because they are able to go back and add artifacts if they would like to keep a detailed record of their learning progress. This allows students to take the information that they have learned and process it in their own way and create an electronic presence that they will have the ability to later reference.
 * Extensions**

· Index Cards · Speakers · Laptops for students/Laptop Cart · Partners handout · [|Ticktacktoe Graphic Organizer] (for each Student) · Handout on domain, range, order-pairs, function and vertical line test · Poster Paper (to record student findings after 3-minute review) · Marker (for students to add their findings to poster) · Overhead Projector/SMART board · Transparencies (of each function) or visuals of each functions (see links below) · Checklist for Understandings Handout (for self-assessment/evaluation) · E-folio Project Handout · Calculator · Text book · Pencil w/ eraser · Assessment Checklist for E-folio Project · Post-it Notes · [|Domain] · [|Range] · [|Ordered Pair] · [|Function] · [|Vertical Line Test] · [|Linear Function] · [|Quadratic Function] · [|Cubic Function] · [|Square Root and Cube Root Functions] · [|Sine, Cosine, and Tangent Functions]
 * __Materials, Resources and Technology__**
 * Materials:**
 * Resources:**

· Laptops (for both teacher and student) · Speakers · GeoGebra, Notebook, Word Processing Software · Calculators · Audio Clips (Created in Garage Band or found in popular songs) · Overhead Projectors/ SMART board · Web Links
 * Technology:**

__Sources For Lesson Plan:__ [|Domain] : This site explains what the domain is, how it is used, and it's role and importance in the use of functions. [|Range] : This link explains what the range is, the purpose it serves, and it's role and importance in conjunction with functions. [|Ordered Pair] : This link explains how an ordered pair relates to a function and how to determine whether a set of ordered pairs is a function or not. [|Function:] This link gives a definition of what a function is, what aspects need to be included for it to be a function, and modeled some situational examples of functions. [|Vertical Line Test] : This link gives a brief summary of what the vertical line test is and how to use it to determine whether or not something is a function. [|Vertical Line Test Activity:] This link helps student practice determining whether or not a graph will pass the vertical line test through connecting points in attempts to create a function. [[http: [] |Linear Function:]] This link gives a great overview on the ways that a linear function can be modeled in an equation and describes how to translate the equation into a graph of the function. [|Quadratic Function] : This link exposes students to the equation of the quadratic function and informs students how to translate the equation into a graph. [|Cubic Function] : This site gives a graphical example of cubic functions and provides the reader with many examples of cubic functions. // [|Cubic Functions] :// This link provides a walk-through on the graphic process of the cubic graph. [|Square Root and Cube Root Functions:] This website explains the equations as well as how to graph both the square root and the cube root functions. [|Sine, Cosine, and Tangent Functions] : Explains the purpose of all three trigonometric functions, their equations, and how to graph each.
 * __Source for Lesson Plan and Research__**

__Pre-Developed Lesson Plans (Research):__ [|Pairing Student's Up:] The teacher in this lesson paired students up and gave them a function based on what they felt their capabilities were. I feel for this lesson, I will pair my students up for the 3-minute reviews and hand the index cards out based on their pre-assessment performances. This way, both students can benefit during the new lesson. [|Refresher] : This teacher decides to use a refresher for the students in the lesson by physically plotting the points or drawing the functions on graph paper. I want to use this idea in my lesson when students come back the second day as a "warm up" exercise. [|Themed Lesson] : This teacher used the various functions to model or relate to their successes in high school. I feel like I could try and tie in the musical component try to promote a bigger picture to my students after seeing how this teacher created a theme for his lesson.


 * __Maine Standards for Initial Teacher Certification and Rationale__**
 * //Standard 3 - Demonstrates knowledge of the diverse ways in which students learn and develop by providing learning opportunities that support their intellectual, physical, emotional, social, and cultural development.//**
 * Rationale**: Students will be asked to consider the task of recognizing functions through an array of approaches for the duration of the lesson on function identification and recognition. To address the "beach ball learner", students will be introduced to the concept of functions through music, personal judgment, peer discussion, lecture and group discussion. Students will be provided with lectures, notes, peer perspectives, online resources, textbook, and feedback in order to absorb all information. This lesson requires students to use new activities to process thinking when it comes to identifying functions and they are able to expand as little or as much on each of the functions as they feel necessary. To address the "clipboard learner", the lesson on functions has been organized so that students are constantly revisiting the purpose of the assignment. When students begin the lesson, they understand that they are going to be working with shapes. As the lesson progresses, students begin develop connections between the sound of the music clip to the shape of the graph, and the graph to the equation. By the end of the lesson, the e-folio provides students with closure on what they have accomplished for the lesson.

To address the "microscope learner" students will be asked to analyze the music clips at the beginning of the lesson. They will have to pay attention to details to match the music clip with the graph and/or equation. Students will also have to be able to discuss their findings with their peers and make justifications on why they decided to include or exclude information on their graphic organizer. To address the "puppy" learner, students will be asked to work together in groups to determine what functions match with which graphs. Throughout the lesson, students are asked to "report out" their findings and progress on representing graphs and the groups are designed so that peers become as support system for others.


 * //• Standard 4 - Plans instruction based upon knowledge of subject matter, students, curriculum goals, and learning and development theory.//**


 * Rationale**: In this lesson, students will gain knowledge on recognizing elementary functions in the equation and graph form. When students are first exposed to the music clips, they may not see the correlation between the clip and the function. As they begin to gain knowledge on the material however, they are going to apply their knowledge back to the music clip and even expand it beyond the music clip. Throughout the lesson, with will be working with the functions in the two forms and going back to the original idea to re-evaluate their original ideas. At the end of the lesson, they will be asked apply what they know by creating an e-folio that is a personal representation of what their knowledge of each of the functions. They will have to apply what they know to create this summative assessment and develop sufficient artifacts that demonstrate their knowledge on the topic at hand. Students will have to use what they know in order to create artifact that are all encompassing of the lesson.

· __Verbal:__ I will be directing students on how to recognize graphs, what the graphs look like, and how to complete tasks all in a verbal manner. · __Logical:__ Students are in a Logical/Mathematical setting in which students need to use mathematics skills to compute answers and process information. · __Visual:__ Graphs of each of the functions are going to be on the screen (produced with GeoGebra) so that students will have a visual demonstration of what each looks like. · __Musical:__ Students will listen to music clips that represent the shape of functions. · __Intrapersonal__: Students work individually to fill out graphic organizer and process the information. · __Interpersonal:__ Students will work in cooperative learning to clear up misconceptions with understanding functions as well as when they revise their thinking about functions.
 * //• Standard 5 - Understands and uses a variety of instructional strategies and appropriate technology to meet students’ needs.//**
 * Rationale**:

Students are going to produce an e-folio at the end of this lesson. The e-folio is a Type II technology because students are given the task of creating evidence of their understandings for a variety of functions. Students will need to create meaningful artifacts that clearly demonstrate their understanding and ability to recognize the material. Students will use GeoGebra, word processing pages and Inspiration to create artifacts that demonstrate their new abilities in function recognition. Students will be asked to create their own examples, so the information becomes embedded in them and their product is reflected in a Type II way. Students will use their personal preferences to create these artifacts and organize them electronically in a way that best makes sense to them. Since it is a Type II technology, students are at an advantage because they are able to go back and add artifacts if they would like to keep a detailed record of their learning progress. This allows students to take the information that they have learned and process it in their own way and create an electronic presence that they will have the ability to later reference.

Throughout the course of the lesson, students were asked to use a variety of formal and informal assessment strategies to evaluate and support the development of the learner. In any lesson, there should be more formative assessment than summative assessment. In this lesson, students use a variety of methods to make their understandings concrete. After students have been "hooked" by the music clips and exposed to the functions, they will use a 3-minute review with a peer to compare graphic organizer and add/change any information. Peer feedback is valuable and helps teach students more about recognizing functions. After they continue to work with the material, students will go back to the "hook" and match the music clip description with the graph of the function. Then, students will be able to return to their graphic organizer to add/change details once the activity is complete. Teacher will provide students will feedback on organizers so that they will be able to produce a good e-folio. Finally, after students complete their lesson on recognizing graphs, they will complete a checklist of their understandings. There will be a space for additional comments where they can write if they have any other questions.
 * //• Standard 8 - Understands and uses a variety of formal and informal assessment strategies to evaluate and support the development of the learner.//**
 * Rationale**:

After the lesson is complete, students will be asked to create an electronic portfolio using Noteshare that demonstrates their understandings on functions. In this e-folio students will be asked to explain and provide examples the terms: domain, range, ordered pair, function, and the vertical line test. Then, students will be asked to explain the shape of the following function and create a graph in Geogebra. They will need to personally create all artifacts and ensure that their e-folio accurately demonstrates their understanding on the topic. This is their summative (or formal) assessment. They will be graded or assessed using a checklist.


 * __Teaching and Learning Sequence__****:**

Lesson one's primary focus is on function recognition and identification. In this lesson, students will be asked to work individually and in groups, so it the classroom arrangement that is best suited for this lesson is the "two's" arrangement. This arrangement is functional because students will not have to rearrange desks in order to do their three-minute reviews and it allows them to collaborate with students several times during the duration of the lesson, which is encouraged. Students will all still be facing the front of the room, which is also an important component because they are being introduced to new ideas. The agenda appears as follows: · Students will receive Ticktacktoe organizer and take seats. (5 min) · Students will listen to music clips. (3 min) · Students will fill in a description of what they listened to in their first column of their graphic organizer (10 min) · Students will receive instruction on [|Domain] (10 min) · Students will receive instruction on [|Range] (10 min) · Students will receive instruction on [|Ordered Pair] (10 min) · Students will receive instruction on [|Function] (10 min) · Students will receive instruction on the vertical line test (10 min) · Students will tie all concepts together (10 min) · Students will be assigned a homework "questionnaire" dealing with the terms that were covered. (2 min)
 * __Day 1:__**

· Students will be handed index cards with a function depicted on the front (mini-hook) and take seats (3 min) · Students will hold a discussion on why functions are important in real life (4 min) · Students will receive instruction on linear functions (7 min) · Students will receive instruction (and do activity) on quadratic functions (10 min) · Students will receive instruction (and do activity) on cubic functions (7 min) · Students will receive instruction (and do activity) on square root and cube root functions (10 min) · Students will receive instruction (and do activity) on sine function (10 min) · Student will receive instruction (and do activity) on cosine function(5 min) · Students will receive instruction (and do activity) on tangent function. (10 min) · Students will find their partner and do the 3-minute review exercise. (5 min) · Students will hand in graphic organizers and I will provide feedback. (2 min) · Students will be assigned textbook homework on how to sketch functions. (2 min)
 * __Day 2__**

· Students will return to their 3-minute review partner and as a class we will revisit the hook (5 min) · Students will be asked to sketch graphs on various functions for practice/go over homework (5 min) · Students will complete checklist of understandings on function identification. (5 min) · Students will be introduced to the e-folio project (instructions and checklists handed out) (10 min) · Students will be shown a demonstration on how to use GeoGebra, Inspiration, and Garageband ( 30 min) · Students will work on e-folio and complete for homework. (25 min)
 * Day 3:**

Throughout this lesson, students will understand that functions can be expressed in a variety of ways. In this lesson particularly, students will be asked to consider the equation and graphical representation of functions and be able to first recognize them and eventually identify them. By understanding the different formats of expression, students are going to be able to better process the upcoming material in the unit. I will expose students to the idea that functions are found everywhere: doctors charts, in measuring how many calories you have burned using exercise equipment, and measuring the pitch of music. By being able to recognize and identify these functions and their expressions in various forms, students are going to be able calculate variables and compute an answer that they many not have been able to arrive the answer to before. This lesson is focused around the Maine Learning Result that **//students understand and interpret the characteristics of functions using graphs, tables, and algebraic techniques.//** In order to initially engage and clearly portray that message that functions are found everywhere, I am going to have them listen to music clips generated in Garage Band and clips taken from songs. Each music clip is intended to represent a different function and students are going to be introduced to functions by being asked to describe what they hear. I will also provide more simplistic versions of music clips where the parabola would be represented by tones instead of beats. I will prepare more than one example for the hook, just in case students do not hear it in the songs. The hook serves as a good resource as a reminder tool. I may also provide images that prompt their memory too and each example will be discussed when we talk about the individual functions (Day 2). Students will be asked to produce the sound that they have heard (with their voices) to describe the graph if they need help keying the tune they need to focus in on.
 * Where, What, Why, Hook, Tailors:** **Intrapersonal, Interpersonal, Musical, Logical/Mathematical**

Students need to become familiar with the definitions of " [|domain] ", " [|range"], " [|ordered pair] ", [|"function] " and the [|vertical line test]. There will be post-it notes under each on of the student's desks and I will ask students to remove the post-it notes and get into the groups that they have been assigned. One group of students will receive cards that say "actors", another group will say "synthesizers", and a third will be "recorders". I will give all students explanation on the terms and provide them with an example of the term. Then, "actors" will have to try and act out the concept, "synthesizers" will have to work together to develop a definition, and recorders will write down what the other two groups have done. Students will rotate roles for each term. After each of the terms has been covered, I will give students a handout on the key points of each of the terms and what each of them means and they will be asked to do an informal blog. The second day of class, students will receive an index card with a function depicted on it. I will return their graphic organizers and students will be able to change or add any of the information that they have included. Then I will begin my instruction on function. I will introduce the students to each of the functions, the equations, and how to identify each of them through the GeoGebra application and having students tell me what I should be looking for in order to determine that it is, in fact, the correct function. Students will be asked to sketch the images by hand and then check their work using the Geogebra software. They will be given an image to relate the function to. This will serve as another means of helping students remember what the graph was looking like. After they have completed this, I will check for understanding in three ways. First, I will have students return to their graphic organizers and correlate the music clip with a function after all information has been presented. Then, a class-wide discussion will be held and students and students will gather a "consensus" on which music clip matched with each functions. I will record the responses on the board. Students will turn to their 3-minute review partner and clear up any misunderstandings that there may be. Then, a new consensus will be taken. Finally, students will have time to go back and change or add information to their graphic organizers. I will provide students with feedback so students will be able to produce a good e-folio.
 * Equip, Explore, Rethink, Revise, Refine Tailors: Bodily/Kinesthetic, Verbal, Interpersonal, Intrapersonal, Logical/Mathematical** (See teacher content notes)

One technique that students are going to use in order to explore the many different components of elementary function expression is by using the ticktacktoe graphic organizer to record vital details of the music clip, the function's graph, and the function's equation. Students will experience function recognition and identification in this lesson through both the lecture as well as the 3-minute review that will give students a chance to discuss the key components of each function. By the end of this lesson, students will be able to recognize and sketch the graphs of basic functions. They will get practice on this throughout the lesson as well as in assigned homework on the first day. Through the hook, the 3-minute lesson plan, holding a group discussion, and having students go back and revise their ideas, they will be able to work on developing their skills to recognize and sketch functions. For the 3-minute review, students will be given index cards at the beginning of class. I will have already pre-grouped students based on skill level because it is essential that students work efficiently and effectively during this time. The index cards have an image of the function. They will need to identify their function and then identify the person who has the same function as them; this is their partner. One student will become an analyzer and work to see if the information on both graphic organizers is accurate and important. The other student will record any differences that need to be addressed. Then, both students will work to determine which information is useful and which information is debatable. Students will remain in their 3-minute review pairs and they will revisit the music clips. In these pairs (or individually), students will use the information that they have collected as well as their homework as a reference to match the music clips with the functions. Then, the 3-minute review groups will report out what they have found. I will record their findings. Any misconceptions will be discussed with other peer groups, and I will probe questions that will lead students in the right direction without giving them the answer. I will reveal the clip representations and which clip was supposed to represent each function. I will allow students to work individually to change/erase/add information to their graphic organizer. I will ask them to turn in what they have produced the first day so I can provide them with feedback and warn them with essential information that they may be missing or points that they may want to consider revisiting.
 * Explore, Experience, Rethink, Revise, Refine, Tailors: Interpersonal, Intrapersonal, Verbal, Logical/Mathematical**

Students will be able to self-asses their progress through a checklist of understandings as well as through the feedback that I provide in the graphic organizers. Will provide timely feedback through verbal comments, written comments on their graphic organizers and blogs, and through addressing questions that I receive from student. When they complete the checklist and are working on their e-folios, I will give students an opportunity to come up and resolve any lingering issues. I will provide times where I will be available for helping students with their e-folio and with any questions that they still do not understand. I will have students upload their e-folio to an artifacts page on a wiki so that students do not have to send electronic files to me. I will give them timely feedback by assessing their product with a checklist in which students earn points for meeting certain criteria. Creating the e-folio and having students sketch the graphs work well because students will be able to go back and refer to these documents for future assignments. The more details that the students are in their e-folio, the more reference material they are going to have in future lessons. The next lesson works more with variations of the elementary/parent functions that we discuss in this lesson, it if they do a good job on this, they will be able to use this to reference different expressions of the same functions in the next lesson.
 * Evaluate, Refine, Tailors: Interpersonal, Intrapersonal, Logical/Mathematical/ Verbal, Visual**


 * Content Notes:**

I have themed my lesson by using the idea from the website [|Themed Lesson]. In this, the instructor gave students a task. In this lesson, students are investigators trying to match music clip with functions.

As students walk into class, I will greet them at the door by handing them a [|Ticktacktoe Graphic Organizer]. After they have received this, they will be asked to take their seats. I will play a series of video clips and they will be asked to respond to each of the clips by describing what they have heard in each of one of the boxes (in the first column of their graphic organizer). We will talk about what each of the sounds indicated a graph would look like as a precursor to the lesson.
 * __Day One:__**

__Domain__ After students have completed that activity, they will be asked to put their graphic organizers aside. Introduce the idea of domain by having students define what the term domain means in everyday life. The definition of domain can be discussed because it can than be applied to it's meaning in the mathematical world. [|Using definitions of domain], here are some of the possible answers that students could come up with:

· sphere: a particular environment or walk of life; "his social sphere is limited"; "it was a closed area of employment"; "he's out of my orbit" · territory over which rule or control is exercised; "his domain extended into Europe"; "he made it the law of the land" · (mathematics) the set of values of the independent variable for which a function is defined · world: people in general; especially a distinctive group of people with some shared interest; "the Western world" · knowledge domain: the content of a particular field of knowledge

After this discussion, I will ask them to think about what domain may mean in the field of mathematics. I will draw the ordered pairs that are listed using the website on [|Domain]. These are the order pairs: {(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}.

I will ask if anyone understands what the domain in in this setting. If a student can answer what the domain might be, I will ask them what an easier way to write this is. I will then use the same set of order pairs and rewrite them in columns of "x" and "y's". I will then separate it a step further and take the set of x values and rewrite them. I will explain that these are the numbers that are considered to be the "domain" or input values, just as they did on the website.


 * domain: **** {2, 3, 4, 6} **

__Range__ After introducing them to the concept of domain, I will ask student to find a relationship that involves a domain, that is, a relationship between x and y and identify x as the domain of the relationship. I will then reinforce the statement that the domain is made up from the set of x values in a function and that the domain is an independent variable. If all students understand this concept, I am going to move on to range.

Using the website for [|Range], it uses the same ordered pair that is used in the domain. I will again ask students what the word "range" means in everyday language. Some of the [|definitions for range] may include:

· scope: an area in which something acts or operates or has power or control: "the range of a supersonic jet"; "a piano has a greater range than the ... ·  change or be different within limits; "Estimates for the losses in the earthquake range as high as $2 billion"; "Interest rates run from 5 to 10 percent"; "The instruments ranged from tuba to cymbals"; "My students range from very bright to dull" ·  the limits within which something can be effective; "range of motion"; "he was beyond the reach of their fire" ·  roll: move about aimlessly or without any destination, often in search of food or employment; "The gypsies roamed the woods"; "roving vagabonds"; "the wandering Jew"; "The cattle roam across the prairie"; "the laborers drift from one town to the next"; "They rolled from town to town" ·  a large tract of grassy open land on which livestock can graze; "they used to drive the cattle across the open range every spring"; "he dreamed of a home on the range" · have a range; be capable of projecting over a certain distance, as of a gun; "This gun ranges over two miles" · a series of hills or mountains; "the valley was between two ranges of hills"; "the plains lay just beyond the mountain range" · range or extend over; occupy a certain area; "The plants straddle the entire state" · a place for shooting (firing or driving) projectiles of various kinds; "the army maintains a missile range in the desert"; "any good golf club will have a range where you can practice" · lay out orderly or logically in a line or as if in a line; "lay out the clothes"; "lay out the arguments" · a variety of different things or activities; "he answered a range of questions"; "he was impressed by the range and diversity of the collection" · crop: feed as in a meadow or pasture; "the herd was grazing" · image: (mathematics) the set of values of the dependent variable for which a function is defined; "the image of f(x) = x^2 is the set of all non-negative real numbers if the domain of the function is the set of all real numbers" · let eat; "range the animals in the prairie" · compass: the limit of capability; "within the compass of education

After students give me some ideas on what they think the word means, I will ask them to consider what it means in mathematics. I will hint to them that the domain and range are related, and see if anyone makes the connection that the range is the set of y-values in the function. The [|Range] website uses the same ordered pairs, so I will return to that example on the board. I will ask student to identify the range in the order pair, in the x-y columns, and then tell me to write out what numbers are in the range. The numbers in the range are :

Range: ** {–3, –1, 3, 6} **

After students arrive to this conclusion, I will ask them to return to their previous example of a relationship between x and y (with their domain already identified), and now determine a range. Then, I am going to ask them to think of another example of a relationship within the classroom and identify the domain and range. They will have to explain their example to the class.

I will then ask them to work out the following problem:

The domain is all values that //x// can take on. The only problem I have with this function is that I cannot have a negative inside the square root. So I'll set the insides greater-than-or-equal-to zero, and solve. The result will be my domain:

–2//x// + 3 __>__ 0 –2//x// __>__ –3 2//x// __<__ 3 // x// __<__ 3/2 = 1.5

Then the domain is "all //x// __<__ 3/2 ".

The graph starts at //y// = 0 and goes down from there. While the graph goes down very slowly, I know that, eventually, I can go as low as I like (by picking an //x// that is sufficiently big). Also, from my experience with graphing, I know that the graph will never start coming back up. Then the range is " //y// __<__ 0 ".

__Ordered Pairs__

Now that students understand what the domain and range of the function are, I will reveal (or remind them) what it means to be an ordered pair. Using the [|Ordered Pair] website, I will explain to them what it means to be an "x" coordinate and a "y" coordinate. Then, I will give them in a pair and show them how to plot an order pair. I will have a coordinate plane mapped out with masking tape on the floor, and I will have students come up and find the following points (from the website)

Graph Ordered Pairs Instructions:

Graph the following ten points: 1. (-2,2) 2. (0,-4) 3. (-4,-4) 4. (0,0) 5. (4,5) 6. (-6,-5) 7. (0,-7) 8. (0,3) 9. (0,6) 10. (0,-8)

I will have my students refine their ideas by talking about order pairs and then going back and identifying the domain and range of the pairs that they have just plotted. They will have to sketch these on graph paper as well.

__Function__ Now that my students have been equipped with all sorts of background information on terms leading up to understanding what is means to be a function, I will ask them reflect on what we have just covered. I will ask for any observations or relationships and record them on the board. Now, I will define what it means to be a function using my [|Function] website. I will use the following information to portray the concept of a function to my students:

A function is a "well-behaved" relation. Just as with members of your own family, some members of the family of pairing relationships are better behaved than other. (Warning: This means that, while all functions are relations, since they pair information, //not// all relations are functions. Functions are a sub-classification of relations.) When we say that a function is "a well-behaved relation", we mean that, given a starting point, we know exactly where to go; given an //x//, we get only and exactly one //y//. Let's return to our relation of your classmates and their heights, and let's suppose that the domain is the set of everybody's heights. Let's suppose that there's a pizza-delivery guy waiting in the hallway. And all the delivery guy knows is that the pizza is for the student in your classroom who is five-foot-five. Now let the guy in. Who does he go to? What if nobody is five-foot-five? What if there are //six// people in the room that are five-five? Do they all have to pay? What if you are five-foot-five? And what if you're out of cash? And allergic to anchovies? Are you still on the hook? Ack! What a mess! The relation "height indicates name" is not well-behaved. It is not a function. Given the relationship (//x//,//y//) = (five-foot-five person, name), there might be six different possibilities for //y// = "name". For a relation to be a function, there must be //only and exactly// one //y// that corresponds to a given //x//.



I will then have my students come up with their own examples of a function (using a relationship and come up with a set of domains and ranges that work. I will come around and individually check their answers so that I ensure everyone understands before moving on.

Using my website on the vertical line test, I will (a) define the terms and (b) give examples of the vertical line test and determining if something is a function or not. The [|Vertical Line Test] definition from the website is: Given the graph of a relation, if you can draw a vertical line that crosses the graph in more than one place, then the relation is not a function.

I will then use the examples of a parabola and a circle and have students evaluate if they are indeed functions are not. There will be additional practice in the following document: [|whatsthatfunctionquestionaire.doc] (see attached) I will supplement my instruction for the day by providing students with the [|What’s the Function.doc] (see attached) handout that briefly describes the general ideas that were discussed in today's class.

Using the completed [|partnersforlesson.doc] (see attached) partners for lessons document, I will hand students index cards that have a function depicted on the front of it. Each student is to go and take their seats and save their index cards for later. I will not answer any questions about functions, just simply ask them to not ask one another about what is depicted on their cards. This is going to serve as a mini-hook regarding today's lesson.
 * __Day 2:__**

Using the [|Linear Function] website, I am going to ask students what they think about when I say the term "Linear function". I will do a quick "brainstorm of ideas" on the board and then have them recall the previous day's music clips. They will be asked to return to their [|Ticktacktoe Graphic Organizer] and match the description of a linear function with a music clip that they heard from the previous day. I will replay any of the clips that they ask me to. I will then put up the equation of y= x. I will show them that this is a linear equation and have them explain to me why a linear equation is a function, without words. I am hoping they will apply what they know about the vertical line test. I will then give them the equation ** Graph ****// y //**** = 7 – 5//x// **. I will have them make a T-chart for this graph (as they did on the website) and then graph the function. I will probe students with questions on why this is similar to the last linear function and how it relates to our discussion on functions.

Next, using my [|Quadratic Function] website, I will write the equation y=x^2 on the board. I will have my students infer what they can tell me about this function by simply looking at the equation. I will then ask them if it is a function. Instead of just drawing the function on the board, I am going to play the music clip that corresponds with the squared function and have my students describe to me what they think it looks like. I will ask them to produce a T-chart and then graph it. They can evaluate if their answer corresponded with the one on the board. I will let them make the mistake of drawing the linear function and then explain to them why it is not in fact a linear function. I will then have them relate the shape to things in their normal world.

Now, using my [|Cubic Function] website, I will deliver instruction on this function. Before even writing anything on the board, I am going to try and prompt my students to see if they can apply what they know about the equation of the squared function and see if the cubed function's equation really looks like y=x^3. I will explain that anything in this form is called a cubic function and that the form x^3 + x^2 - x + 4 in no way disqualifies it from being a cubic function. I will then have my students create a T-chart. I will have them produce a graph and ask them what some relationships involving y^3 might be.

Next, I am going to deliver instruction on square root and cube root functions using the [|Square Root and Cube Root Functions] website. I will show them what the equations and general forms look like then use the following information to reinforce the ideas:



Now, I will ask them how these graphs relate to those of x^2 and x^3. I will play the music clips again for them.

Finally, I will use the [|Sine, Cosine, and Tangent Functions] as my last class example. I will use the following explanation to help students understand sine, cosine and tangent functions.

At first, trig ratios related only to right triangles. Then you learned how to find ratios for any angle, using all four [|quadrants]. Then you learned about the [|unit circle], in which the value of the hypotenuse was always //r// =1 so that //sin//(θ)= //y// and //cos//(θ) = //x//. In other words, you progressed from geometrical figures to a situation in which there was just one input (one angle measure, instead of three sides and an angle) leading to one output (the value of the trig ratio). And this kind of relationship can be turned into a [|function]. Looking at the sine ratio in the four quadrants, we can take the input (the angle measure θ), "unwind" this from the unit circle, and put it on the horizontal axis of a standard graph in the //x,y//- [|plane]. Then we can take the output (the value of //sin//(θ) = //y//) and use this value as the height of the function. Reserved

As you can see, the height of the red line, being the value of //sin//(θ) = //y//, is the same in each graph. In the unit circle on the left, the angle is indicated by the green line. On the "regular" graph on the right, the angle is indicated by the scale on the horizontal axis. If the green angle line had gone backwards, counting into negative angle measures, the horizontal graph on the right would have extended back to the left of zero. If, instead of starting over again at zero for every revolution on the unit circle, we'd counted up higher angles, then the horizontal graph on the right would have continued, up and down, over and over again, past 2π.

**// The Sine Wave //**

From the above graph, showing the sine function from –3π to +5π, you can probably guess why this graph is called the sine "wave": the circle's angles repeat themselves with every revolution, so the sine's values repeat themselves with every length of 2π, and the resulting curve is a wave, forever repeating the same up-and-down wave. (My horizontal axis is labelled with decimal approximations of πbecause that's all my grapher can handle. When you hand-draw graphs, use the exact values: π, 2π,π/2, etc.) When you do your sine graphs, don't try to plot loads of points. Instead, note the "important" points. The sine wave is at zero (that is, on the //x//-axis) at //x// =0, π, and 2π; it is at 1 when //x//= π/2; it is at –1when //x// = 3π/2. Plot these five points, and then fill in the curve. We can do the same sort of function conversion with the cosine ratio:

The relationship is a little harder to see here, because the unit circle's line is horizontal while the standard graph's line is vertical, but you can see how those two purple lines are the same length, while the angle measure is moving from zero to 2π. And just as with the sine graph, the cosine graph can be extended outside the interval from zero to 2π:

**// The Cosine Wave //**

As you can see from the extended sine and cosine graphs, each curve repeats itself regularly. This trait is called "periodicity", because there is a "period" over which the curve repeats itself over and over. The length of the period for the sine and cosine curves is clearly 2π: "once around" a circle. Also, each of sine and cosine vary back and forth between –1 and +1. The curves go one unit above and below their midlines (here, the //x//-axis). This value of "1" is called the "amplitude". When you graph, don't try to plot loads of points. Note that the cosine is at //y// =1 when //x//= 0 and 2π; at //y// =0 for //x//= π/2 and 3π/2, and at //y// =–1 for //x//= π. Plot these five "interesting" points, and then fill in the curve.

I will draw the graph of tangent as well, but this one is usually not used in graph form because it does not give any information. Basically I am going to use the website for concepts and to be able to graph sine and cosine accurately.

Students will go back and review their [|Ticktacktoe Graphic Organizer] and I will replay all music clips. Students will get into their three-minute review pairs and fill out their graphic organizers and hand them in for feedback. After their three minute review session, they will have to complete the [|whatsthefunctionchecklist.doc]. Then, students will be given the sketching graphs [|sketchinggraphs.doc] document to review for homework.

Today, I am going to provide my student with laptops. I will give them a quick tutorial on how to use [|GeoGebra Tutorial (How To's)], such as how to enter the equation of a line, identify points on a graph, and take a screen shot of the image. Students will also have a access to this link.
 * __Day 3 :__**

Then, I will give them a quick tour around Garageband by using my [|Garageband Tutorial (How To's)]. Students are required to include sound clips into their e-folios, so they will need to know how to create a sound clip.

Inspiration is optional, but I will let them play around with the webbing process for five minutes or so by having them re-create their graphic organizers. I will use the [|Inspiration Tutorial (How To's)] to show students all of the features that Inspiration has to offer and how to navigate around the program.

Lastly, I will use the Noteshare Tutorial (Video) to have students learn how to set up their e-folio. This is the most efficent way to project the same information to every student.

Students will receive the following two documents: [|whatthefunctionefolio.doc] [|functionsefoliochecklist.doc]

They will be given the rest of the class period (and until the next class) to ask questions and complete the assignment. It will be graded using the checklist found attached.

Other Teacher's Lesson Plans:

[|Refresher][|Pairing Student's Up]

Other Resources that students may need:

[|Ticktacktoe Graphic Organizer] [|What’s the Function.doc] [|whatsthatfunctionquestionaire.doc] [|sketchinggraphs.doc] [|whatsthefunctionchecklist.doc] [|whatthefunctionefolio.doc] [|functionsefoliochecklist.doc] Any of the visuals or audio provided by teacher will be made available to students on class wiki.
 * Handouts (see attached)**
 * [|partnersforlesson.doc] **


 * Reflection:**