L3+Emery,+Jordan

COLLEGE OF EDUCATION, HEALTH AND REHABILITATION LESSON PLAN FORMAT**
 * UNIVERSITY OF MAINE AT FARMINGTON


 * __Teacher’s Name__****:** Ms. Emery ** __Date of Lesson__:** Lesson 3 (Interpretation)
 * __Grade Level__****: **10 ** __Topic__:** Function

Students will understand that there are many techniques used to solve functions and change the appearance of functions. Students will know the techniques horizontal stretch, vertical stretch, reflection over x-axis, reflection over y-axis. Students will be able to translate functions of graphs according to equation manipulations.
 * __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 learn to recognize the graphs of functions that have been rotated or have had minor transformations performed to them.

There are informally assessed on their learning using a variety of methods. First, students are pre-assessed and begin their learning process using a [|KWL] (What I know, what I Want to know, what I've learned) [|graphic organizer]. They will be asked to fill in their first two columns. Then, after some instruction, they will be asked to revisit this graphic organizer and highlight the answers that they feel were answered during the instruction. They will also have the opportunity to write questions that had not been considered when the lesson first started.
 * __Assessment__**
 * Formative (Assessment for Learning)**

Aside from just recording their answers, another way that students are informally assessed is by reflecting on their prior knowledge and recollections of familiar function. For example, students will be asked to think about how the ordered pairs are affected by graph transformations. Students will also be asked to think about what a graph would look like if it were reflected across the y=x line. Also, after some instruction and activities have taken place, students will be asked to list the big ideas that have been covered in class (as a group). They will have a chance to fill in the last section of the graphic organizer and hand it into me for some constructive comments (but not grades). This will allow me to clear up any misconceptions and allow me to know if I have to revisit any topics before moving on. Another way that students are formatively assessed is through working in small groups to quickly engineer a model to explain a reoccurring misconception and explain it to the class. I will probe the students in the right direction in order to explain the idea to the class, but not give them the answer. Students are given time to self-assess their learning through a blog entry.

The product that students will need to create is an audio recording/ podcast using Garage Band. Students will be asked to create a five minute podcast explains each of the graph transformations. In their podcast, they will need to create an informational segment that covers 1) a horizontal stretch, 2) a vertical stretch, 3) a reflection over the x-axis, and 4) a reflection over the y-axis. In this segment, students should address how different functions react differently to these transformations as well as how the domain and range change. Music that is being played while speaking should reflect the affects of the graphical transformation. They will be graded using a rubric.
 * Summative (Assessment of Learning)**

__Technology__: Students are asked to use technology to help increase their understanding on transformations. In the beginning of the lesson, they are asked to use the computer to play a " [|5 difference game] " on the computer. Then, they will be asked to consider images that have been reflected, rotated, and transformed using GeoGebra. Students will also be asked to use Garage Band to create their final product (a podcast) for this lesson.
 * __Integration__**

__English:__ Students will be asked to use their descriptive writing skills to develop a podcast that describes the various transformations of functions.

__Physical Education:__ Students will be asked to get up and perform activities to help them understand the terms that I will “equip” them with in this lesson.

__Speech__: Students will use their speaking skills to produce a good podcast for the group to listen to.

Using "Round Robin Brainstorming", students will work together to add to their "W" column and create a list for the "H" column. Students will be broken up into groups by the order that they finished the 5 differences game. The first four that finished will make up the first group, the next fur will make up the second group, and so on. When they are in the group, one student will become the designated "recorder". The other students will address the "W (what I want to know)" column while the "recorder" records this information. Half way through this discussion, someone else will become the recorder and the discussion will continue. Then, students will switch recorders again and address the "H (what I have learned)" column. Students will switch recorders half way through this discussion so that all four people have been the recorder while the other three have been discussers for this brainstorming session. During this time, I will be walking around and listening to the conversations and trying to spark students interest and conversation if there is little conversation going on in the group.
 * __Groupings__**

__Verbal:__ Students will be in a classroom setting where explanations, descriptions, and instructions are verbal. Students need to listen to teachers and peers for key information. __Logical/Mathematical:__ Students are asked to use their knowledge of functions and apply their knowledge of their shapes and how a number or negative sign will affect the appearance of the graph. __Bodily/Kinesthetic:__ Students will act out the terms that they need to be “equipped” with. __Visual:__ Students will be exposed to "before" and "after" pictures of graphs so that students can visualize the affects transformations have on graphs. __Musical:__ Students are asked to create a Garage Band product that entails the information learned in the lesson. They are also to include their own interpretation of musical symbolization of the newly transformed functions. __Intrapersonal:__ Students are asked to work individually to complete the tasks of filling out the two first columns of the graphic organizer as well as revisiting the organizer to assess their personal learning growth. __Interpersonal__: Students are asked to work in groups to complete the graphic organizer as well as to refine some topics that are unclear. I will review student’s IEP, 504 or ELLIDEP and make appropriate modifications and accommodations.
 * __Differentiated Instruction__**
 * Strategies:**
 * Modifications/Accommodations**

Absent students will be prompted to refer to the class wiki and gather as much information that the can from the wiki. They will also be asked to gather any notes or information that has been taken in class from their absent buddies that have been predetermined since the beginning of the year. Once students return to school, I will ask them to come see me if they are still unclear with any of the material and to work with me on how to fill out the graphic organizer and grasp any concepts that peers or the wiki could not explain. If a student has missed an entire lesson, I will have this student listen to some examples of podcasts to help him learn the material on function transformations (and then demonstrate to my students how their homework is beneficial).

Students will need to create a podcast at the end of this lesson. In order to do this, I will need to get them inspired by giving them an authentic assessment task. In order to get the students involved in creating their own podcasts about graph transformations, they will be inspired by listening to a Grammar Girl podcast. I will then ask them to imagine themselves as "Math Man" or "Graph Girl", and they need to create a podcast of their own. They will use Garage Band, as well as their knowledge on graph transformations, to create an engaging and informational podcast. They will be encouraged to use sound files that were created in earlier assessments to explain the graph transformations. This is a Type II use of technology because students have to not only record themselves explaining graph transformations; they have to create an entertaining and informational broadcast that is all-encompassing and captures the big ideas of the unit and speaks to the listeners. Every person's podcast will be unique and explain the transformations in a different way, so it provides a good index for someone who still does not fully grasp the concept.
 * Extensions**

__Materials:__ · Laptops · Projector · SMART Board · TI-84 Calculator Program · Whiteboard/markers · [|KWL graphic organizer] · [|5 difference game] · GeoGebra Images (the shapes of graphs) · Cardboard cutouts of different graphs and their transformations · Graph Transformation Worksheet (Activity Option worksheet) [|transformationactivity.doc] · Transformations "Cheat Sheet" Table [|transformationcheatsheet.doc] · Grammar Girl Podcast · Math Man or Graph Girl Podcast Descriptor Handout [|podcastexplanation.doc] · Podcast Rubric [|podcastrubric.xls] · Washable Markers · Poster paper · microphones (to record) · Garage Band · Calculators
 * __Materials, Resources and Technology__**

__Resources:__ · [|Getting the idea: An introduction to horizontal stretching and shrinking] · [|Horizontal Stretch and Shrinking of Parabolas] · [|Transformations of graphs walk-through] · [|Vertical and Horizontal Stretches of Trig Graphs] · [|All-inclusive website of graph transformations] · [|Graph transformations and reflections] · [|Horizontal and Vertical Graph Transformation Videos] · [|Transformation Practice Problems] · [|Interactive Transformation Practice] · [|Transformations on a Graphing Calculator Lesson Plan] · [|Transformation Lesson Plan]

__Technology__: · Laptops · Grammar Girl Clip · Garage Band Clip · Microphones · Projector · GeoGebra Images · Calculators

· [|Getting the idea: An introduction to horizontal stretching and shrinking] : This website introduces the reader to the idea of horizontal and vertical movement as well as stretching and shrinking (as well as provides pictures). This website is more of a supplemental website if a student needs another resource to look at information, because there are better websites that explain these ideas much more thoroughly. This website is consistent with using parabolas though, and this may be good for one to consider. · [|Horizontal Stretch and Shrinking of Parabolas] : This site works solely with parabolas to ensure that students get the concept of horizontal and vertical transformations. The quiz at the end provides them with a quick and easy self-assessment that allows them to see if they need to practice their skills further. · [|Transformations of graphs walk-through] : This site provides an excellent slide show that walks through what a transformation is, what kinds there are, how they are used and includes examples. I plan to use this website for much of my lesson. · [|Vertical and Horizontal Stretches of Trig Graphs] : This website give students another medium of uniformity to think about horizontal and vertical transformations in. · [|All-inclusive website of graph transformations] : A great website that provides visuals as well as a verbal explanation of each sort of graph transformation that is included in my lesson. · [|Graph transformations and reflections] A very generalized explanation of graph transformations. Not very student friendly if they are not math-lovers, but is sufficient for content notes. · [|Horizontal and Vertical Graph Transformation Videos] : This website is a good way for students to gain extra perspective on vertical and horizontal graph transformations. It may also make a good model for their podcasts (if they just listen and not watch) to see what parts need to be explained more specifically if they do not have a visual. · [|Transformation Practice Problems] : This website provides an explantation for each of the graph transformation as well as provides great practice problems to have students identify what transformations are occurring. · [|Interactive Transformation Practice] : If you scroll down the page to "Graph Transformations", there are several links that help students interact and visualize what each transformation does to the graph. · [|Garage Band Refresher:] : This link provides students with a resource if they get stuck using Garage Band on their own. · [|Transformations on a Graphing Calculator Lesson Plan] : This teacher is using calculators to show the difference between the translated graph and the parent function. I had not considered doing this, but now realize that it is not a bad idea. · [|Transformation Lesson Plan] : In this lesson plan, the teacher gives the students two functions and has them describe what was done to the function to make it look the way that it does. I think that this is a good exercise and will be good practice for my students when they go and do their podcasts. This will allow me to assess their description skills.
 * __Source for Lesson Plan and Research__**
 * Lesson Plan Sources:**
 * Research**:


 * __Maine Standards for Initial Teacher Certification and Rationale__**
 * //Standard 3 - Demonstrates a 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://** The varied techniques and approaches that this lesson provides demonstrates a knowledge of the diverse ways in which students learn. This lesson provides students will many different hands on activities, visuals, and verbal instruction in order to understand the concept of graph transformations. After students have tried all the methods of thinking about graph transformations, they will be able to fill out their activity option worksheet whichever way they feel they can best learn the information. This gives students a sense of personal freedom and allows them to explore a variety of resources.

This lesson also provides other students with a sense of structure and organization. The lesson is taught in such a sequence that students will be able to successfully complete the podcast easily and flawlessly by the end of the lesson. First, students will be given time and a graphic organizer to explore the many different ways that a graph can be transformed. They will have to pay attention to details in order to fill in their "Transformation Cheat Sheet" table. Once they are given the many different tools (and ways to think about graph transformation), including online resources, calculators, and cardboard cutouts, they will be able to do the worksheet based on what they have learned. The podcast is prompted by an abstract and a rubric (and modeled by a Grammar Girl podcast), so there are clear expectations on what is expected for the podcast. The podcast also provides for clear closure and a summative assessment for a very organizational and structure orientated student.

The students that feel most comfortable with discovery learning will feel most accompanied when being introduced to the many different ways to think about graph transformations. Graph transformations are very location specific, so their keen eye for detail will work in their favor for the entire lesson. They will be able to assist those students who struggle to grasp concepts on location and paying attention to how small changes (such as the placement of a two) could significantly alter the shape of a graph. These students will feel comfortable in any situation because all of the activities in this lesson are detail specific.

The student who needs a comfortable and encouraging atmosphere will feel very comfortable during this lesson because students will be asked to work collaboratively in order to make progress in this lesson. Students will be encouraged by one another to stay on topic and get creative to come up with questions because it will only benefit them later in the lesson. The fact that there are multiple ways to think about this concept provides for an encouraging atmosphere because this sends the message that not all students are going to understand this concept the same way. These students will be able to group up and work together to explain and discuss their thought process on each of the graph transformations within the way that they have chosen to consider graph transformations. Since I will be only answering questions and providing guidance, not answers, students will feel prone to listen to one another and become support groups to help one another understand what is going on.


 * //Standard 4 - Plans instruction based upon knowledge of subject matter, students, curriculum goals, and learning and development theory.//**
 * //Rationale://** In this lesson, students are asked to consider graph transformations and the many images that are created from these graphs. Not only are they asked to learn what happens when something is multiplied by a factor of two or three is subtracted, but they are asked to apply it. They are going to be asked to examine the several graph transformations that occur and not only understand what they look like, but be able to explain them without a visual. They will need to apply what they know to human experience to make their listener understand what is happening to the graph and why it is important to know this. The podcast itself will demonstrate what makes sense to the student and how well they can explain the concept of graph transformations to another individual.

· __Verbal:__ Students will be in a classroom setting where explanations, descriptions, and instructions are verbal. Students need to listen to teachers and peers for key information. · __Logical/Mathematical:__ Students are asked to use their knowledge of functions and apply their knowledge of their shapes and how a number or negative sign will affect the appearance of the graph. · __Visual:__ Students will be exposed to "before" and "after" pictures of graphs so that students can visualize the affects transformations have on graphs. · __Musical:__ Students are asked to create a Garage Band product that entails the information learned in the lesson. They are also to include their own interprertation of musical symbolization of the newly transformed functions. · __Intrapersonal:__ Students are asked to work individually to complete the tasks of filling out the two first columns of the graphic organizer as well as revisiting the organizer to assess their personal learning growth. · __Interpersonal__: Students are asked to work in groups to complete the graphic organizer as well as to refine some topics that are unclear.
 * //Standard 5 - Understands and uses a variety of instructional strategies and appropriate technology to meet students’ needs.//**
 * //Rationale://**

Students will need to create a podcast at the end of this lesson. In order to do this, I will need to get them inspired by giving them an authentic assessment task. In order to get the students involved in creating their own podcasts about graph transformations, they will be inspired by listening to a Grammar Girl podcast. I will then ask them to imagine themselves as "Math Man" or "Graph Girl", and they need to create a podcast of their own. They will use Garage Band, as well as their knowledge on graph transformations, to create an engaging and informational podcast. They will be encouraged to use sound files that were created in earlier assessments to explain the graph transformations. This is a Type II use of technology because students have to not only record themselves explaining graph transformations; they have to create an entertaining and informational broadcast that is all-encompassing and captures the big ideas of the unit and speaks to the listeners. Every person's podcast will be unique and explain the transformations in a different way, so it provides a good index for someone who still does not fully grasp the concept.


 * //Standard 8 - Understands and uses a variety of formal and informal assessment strategies to evaluate and support the development of the learner.//**
 * //Rationale://** Students are asked to demonstrate their understanding of graph transformations in a variety of way. When the [|KWL graphic organizer] is handed out, they will be asked to fill out the first column. Then, students will create a poster list in groups of 4 and present to the class what they already know. This serves as a pre-assessment for the lesson. Then, students will be exposed to a variety of informal assessment strategies that allow me to informally assess and track their learning progress and understanding throughout the lesson. Students are asked to discuss how ordered pairs are affected by graph transformations as class. I will be able to give them immediate feedback from this exercise. Then, I will track their progress informally by having them use Round Robin Brainstorming to fill in the remainder of their [|KWL graphic organizer] . In this exercise, I will be able to track progress. The graphic organizer itself acts an in informal assessment. When students are asked consider a variety of methods on how to think about solving problems, it will also give me an indication on who is catching on and who is not.

Students will be formally and authentically assessed using a podcast and a rubric. This will take place instead of a quiz or a test over the topic. The product that students will need to create is an audio recording/ podcast using Garage Band. Students will be asked to create a five-minute (or longer) podcast that explains each of the graph transformations. In their podcast, they will need to create an informational segment that covers 1) a horizontal stretch, 2) a vertical stretch, 3) a reflection over the x-axis, and 4) a reflection over the y-axis. In this segment, students should address how different functions react differently to these transformations as well as how the domain and range change. Music that is being played while speaking should reflect the affects of the graphical transformation. They will have to consider the role of "Math Man" or "Graph Girl" and model the format of "Grammar Girl" or another informative podcast. Their audience is a group of listeners that struggle to understand the basic graph transformations and are not mathematically inclined people. They will be graded using a rubric.


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

For this lesson, students are going to be introduced to the lesson in the mindset that they are going to become a Math Spokesperson/ podcast idol just as Grammar Girl has. Throughout the lesson, they will be reminded that they need to refine and be as familiar with the concept of graph transformations as possible because they are, in fact, going to have to produce a podcast of their very own at the end of their lesson. For this lesson, students will be seated in a perimeter to mimic a large "expert" conference to discuss transformations. This set up is most functional for the simulation of the authentic assessment and applied learning, and it can be easily manipulated to work in our cooperative learning groups (Round Robing Brainstorming). The third day, the room will be arranged in clusters so that students that choose to work on the worksheet a particular way can sit together. The lesson agenda looks as follows:

__Day 1__**:** · Students will enter room, log onto laptop, and go to wiki (5 min) · Students will listen to the "Grammar Girl" podcast and be introduced to their end of the lesson task (**both handouts will be given**) (5 min) · Students will play the [|5 difference game] (log how long it took them to do it) and consider the two images projected on the board(5 min) · Students will be handed out the [|KWL graphic organizer] and asked to complete the "K" column with the person sitting to the left of them. (7 min) · Students will discuss what they know as a class and any prior misconceptions will be addressed at this time (10 min) · Students will be introduced and do the calculator activity with horizontal and vertical shifts (10 min) · Students will be introduced to and do a calculator activity with a vertical stretch (15 min) · Students will be introduced to and do the calculator activity with a horizontal stretch. (10 min) · Students will receive their **Transformations** **Cheat Sheet Table** (10 min) · Students will hand in their graphic organizers for comments (feedback) and students will be asked to investigate the remaining concepts on their "cheat sheet table" for homework (1 min)

__Day 2:__ · Students will enter room, select laptop, return to wiki (5 min) · I will return graphic organizers for students (During above 5 min) · I will review yesterday's concepts using cardboard cutouts of graphs (and their translated images) and I will have students match the equations with the graphs (10 min) · Students will be introduced to and do the calculator activity with a reflection over the x axis (15 min) · Students will be introduced and do the calculator activity with a reflection over the y axis (10 min) · Students will be asked to get into groups according to how fast they finished their 5 differences game (2 min) · Students will use Round Robin Brainstorming to complete the last two columns of their graphic organizer (15 min) · Students will be given two functions and try to figure out what the transformations are (5 min) · Students will be given time to fill in their cheat sheets as a group (5 min) · They will turn in their cheat sheets · Students will use the rest of the time to post a blog entry of what they have learned in today's class (7 min) · Students will be asked to start work on their podcast scripts for homework and we will review the abstract and rubric (5min)

__Day 3:__ · Students will be asked to rearrange the room in "clusters" (2 min) · Students will enter room, log onto laptop, and go to wiki. In the meantime, they will receive their graphic organizer with more feedback and their cheat sheets. (5 minutes) · Students will be handed out the **Graph Transformation Worksheet** and asks students to complete it using (a) calculators, (b) by the video, (c) by using resources online, (d) by using the cardboard cutouts. (20 min) · As a class, we will make a list of the "big ideas" that have been covered in the lesson (10 min) · Students will listen to a clip of the [|Horizontal and Vertical Graph Transformation Video] to see how they may need to explain their transformation (3 min) · Students will have a quick refresher on [|Garage Band] (10 min) · Students will have the rest of the period to work on their podcasts (30 min)

In this lesson, students will understand that graphs take different forms and they can appear very different when there are transformations performed on them. They will understand that there are many ways that functions can be manipulated and it causes the shape and solving of the graph to be different. Students need to learn this content because it happens in our everyday life. Whenever we look in a mirror, go up a step, move over to let someone by, or do a cartwheel, we are applying a transformation. The same thing happens with graphs, with the addition, subtraction, multiplication, and division of numbers. These transformations are vital because graphs are not going to be in their parent form, and in order to model some of the data that is to be looked at, these transformations need to be understood. This lesson meets the Maine Learning Result requirement that **//allows students to understand and interpret the characteristics of functions using graphs, tables, and algebraic techniques//**. In this lesson students are hooked in two different ways. First, they are hooked when they are presented with final product task and the role that they are going to have to take on. If that does not get them hooked, they are going to have the opportunity to play a 5 differences game and asked to infer what they think this has to do with graph transformations.
 * Where, What, Why, Hook, Tailors:** **Interpersonal, Verbal, Logical/Mathematical, Musical, Visual**

For this lesson, students will need to know the techniques horizontal stretch, vertical stretch, reflection over x-axis, reflection over y-axis. Before beginning, they will fill out the "K" in their KWL graphic organizer. I will ease into this by giving them something that they are familiar with: vertical and horizontal shifts. I will give my students a non-math example of a vertical shift by stepping up on a desk. I will ask them how I have moved. I will do the same thing to describe the horizontal shift. I will then ask my students to explain how they would do this mathematically. I will have them suggest ideas and we will record them on the SMART board. Then, using the TI-84 graphing calculator program, I will ask student to follow along as we try out our ideas. From the [|Getting the idea: An introduction to horizontal stretching and shrinking] link, I will use the example of a parabola that they have translated up one unit. I will do problems 1 and 2 with them right on the SMART board. Then, I will use the [|All-inclusive website of graph transformations] website to attain my explanations and generalizations for each of the graph transformations that we have talked about. This website provides an example for each, so I will use the same calculator activity to strengthen their understanding of graph transformations. I will have them do everything by hand first, and then do it on the calculator. (**See content notes)**. Then they will fill out the last two columns of the KWL graphic organizer by using Round Robin Brainstorming after they have received all of their information. After this activity, students will see the same information again using cardboard cutouts of different functions. This will appeal to the bodily/kinesthetic learners of the group. The floor will act as the coordinate plane. After this, students will be asked to fill in their graphic organizer. If there is a misconception, this is where students will work in small groups to clear up misconceptions. Then, they can refine their ideas by using the wiki space to explore an index of websites that explain graph transformations in yet another way. The next day, I will act as a "coach" to help my students solidify ideas. Students will work collectively to coordinate a list of the big ideas that were covered in the lesson. Then I will use the SMART board to help refresh my students on how to use Garage. I will have the Grammar Girl podcast available for model and I will allow discussion during this time period. (**See content notes for specifics on instruction**).


 * Equip, Explore, Rethink, Revise, Refine, Tailors:** **Intrapersonal, Interpersonal, Bodily/Kinesthetic, Musical/ Verbal, Visual, Logical/Mathematical**

In this lesson, students explore their learning and organize their thoughts by using the KWL graphic organizer. At the beginning of the lesson, they are asked to complete the "K" (What I know) section of the graphic organizer in pairs. They are then to report out what they know to the class. They are then asked to fill out what they want to know and what they have learned during the middle of the lesson. A refined version of the last column is their " Graph Transformation Cheat Sheet Table". This is a finalized version of what they know and can serve as a reference sheet when they make their podcast. Students experience this lesson through filling out the last two columns using Round Robin Brainstorming as well as when I allow them to fill out the activity worksheet in whatever way they want. This allows them an option to experience the graph transformation any way that they want. In this lesson, they are asked to interpret the graph transformations, apply them to real life, and express the graph transformations in a different way. This is shown and demonstrated during instruction as well as the fact that I am asking them to create a podcast with no visuals that will reach an audience that typically does not understand math. This will provide the students with the skills that will allow them to use the skills of identifying and translating graphs according to manipulations. In this lesson, students will be grouped by how fast they finish their 5 differences game. The top four will be grouped together, and so on. Since there is no correlation between mathematical knowledge and being able to spot differences, I figured that this was a fairly ambiguous way to group students. Once they are in groups, one student will become a recorder. Others will be "discussers" and try to offer as many pieces of information for the recorder to write down. The role of recorder will switch until all students have the opportunity to attain that position. They will show evidence of learning through group discussion and through their graphic organizers. Their big assessment is their summative assessment, which is a podcast. Before they take on that task, they have several chances to rethink, revise, and refine their work. They will rethink graph transformations by comparing them transformations of ordered pairs. They will revise their learning by listing the big ideas as a class on the board. Finally, they will refine their ideas by clearing up misconceptions by using cardboard cutouts.
 * Explore, Experience, Rethink, Revise, Refine, Tailors:** **Interpersonal, Intrapersonal, Mathematical/Logical, Verbal, Visual**

In this lesson, students will have the opportunity to self-assess in several ways. They get to self- assess their learning from the moment they receive their graphic organizer and filling in what they already know. By doing this, they are already gauging where they stand in the lesson. Next, when they fill in their "Graph Transformation Cheat Sheets" they will be able to see how much they have learned and what they still need some more information on. They get to assess themselves further when discussing the different topics in class (such as the big ideas, what they know, etc). This will give them a perspective on where they stand and what they need to catch up on. I will provide timely feedback by answering questions in class, prompting clear problems as a group, and collecting graphic organizers and cheat sheets and writing comments after each day. The class discussions are my best advantage to provide feedback for students because I can reach the broadest spectrum of students. The homework assignments that I assign will help ease the stress of creating the podcast. This connects to the next lesson because they will have to solve functions that have these transformations applied to them.
 * Evaluate, Refine, Tailors: Interpersonal, Intrapersonal, Verbal, Visual**

__Grammar Girl:__ Students will be asked to listen to a [|Grammar Girl Episode] to get a feel for what they are going to have to complete for a task by the end of the assignment. This episode is on comparing two words, so I think it works nicely because students will have to discuss horizontal and vertical components. Examine Grammar Girl's style, voice, flow, and background music to the class to that they understand the elements of the their graph
 * __Content Notes__**
 * Explain that they can choose to do a segment either over horizontal and vertical transformations or x and y axis reflections

__5 Differences Game:__ Have students log onto their laptops. On the wiki page, there will be a link the reads " [|5 difference game] ". Have students click the link, but do not allow them to start to find the difference until time is being kept track of (using the stopwatch feature on the SMART board). After, have them consider this image:



After they identify it as a quadratic function, I will ask them to consider this picture:

I will ask how the 5 difference game relates to this exercise. I will ask them if they can identify what has happen the functions. Then I will show them the equations. The red line is: y= x² The blue line is: g(x) = -2 (x + 2)² - 1

- Students will be handed the [|KWL graphic organizer] during Day 1. The first column that they are asked to complete could vary with information.

- The class discussion that is based from the "K" column of the graphic organizer is heavily based on what students produce. I am anticipating vertical and horizontal shifts as well as the shapes of graphs. Refer to Lesson 2 wiki resources for explanations on different graphs.

- Next, I will begin my discussion on graph transformations. The lecture should begin with vertical shifts, because this is the easiest one for students to identity. I will stand up on a chair and ask them how my position has changed. I will do it again by standing on a chair. I will as them if I have changed shape at all (they should answer no, only my position has changed). Then, using the [|Getting the idea: An introduction to horizontal stretching and shrinking] link, I will use the example of a parabola that they have translated up one unit. I will do problems 1 and 2 with them right on the SMART board. will go through the same process to explain horizontal shift. I**t is important point out that (x + k) move the function left, and (x-k) moves the functions** right. These are the problems:
 * Vertical Shift and Horizontal Shift:**


 * When you alter a graph, you //transform// it. If you transform a graph without changing its shape, you //translate// it. Vertical and horizontal transformations are translations. When //y = f(x) + d//, shift (translate) the graph of //y = f(x)// vertically (upward if //d > 0//, downward if //d < 0//). Example:**


 * 1. Problem: Translate //y = x2// upward by //1//.**

graph upward //1//. Rewrite the equation to do this, and then graph.
 * Solution:** You have been asked to shift the

//y = x2 + 1//

The figure below is a graph of the solution. 


 * When //y = f(x + c)//, translate the graph of //y = f(x)// horizontally (left if //c > 0//, right if //c < 0//). Example:**
 * 2. Problem: Sketch the graph of //y = |x + 2|//.**

shift it to the left 2 places.
 * Solution:** First, graph //y = |x|//, then

The figure below is a graph of the solution. 

I will then write that a **vertical transformation is** **y= x+ k or y= x- k** to generalize the idea for my students.

I will use a parabola (Pages 2 and 3) from [|Transformations of graphs walk-through] to give them more practice with this idea. I show them a picture of the graph y= x². Then, I will show them a picture of y= [x+3]² and they will have to come up with the equation. Again, will show them a photo of y= [x-5]². They will have to produce the equation. I will vertically translate units as well by drawing pictures and having them translate it. Then, I will draw a picture of y= [x-4]² + 2, and students will have to come up with this equation.

Points I want to highlight: In order to vertically transform a function, a number will be added (or subtracted) at the end of the original equation In order to horizontally transform a function, x+a number (or x-a number) will be substituted for x in the original equation (Example: f(x)=x2 will be shifted left by 3 for the equation f(x)=(x+3)2

Using the [|Transformations of graphs walk-through] website again, I will explain horizontal stretch and shrink. __Points I want to include:__ - Graphs of functions may also be vertically shrunk and stretched by multiplying the y-coordinate by a number. - The x-coordinate will not change. - To vertically stretch a graph, multiply the entire function by a number greater than 1. - To vertically shrink a graph, multiply the entire function by a number less than 1.
 * Vertical and Horizontal Stretch and Shrink:**

Then I will use the following examples on the board. I will have my students graph these equations by hand by creating tables then plotting points. Then I will have them enter them into their calculators and compare the magnitude of the graphs. This is from the [|Transformations of graphs walk-through] website.

Original function f(x)=(x-1)2+2 black b Vertical stretch (multiply by 3): f(x)=3[(x-1)2+2] red b Vertical shrink (multiply by .5): f(x)=.5[(x-1)2+2] green


 * __Horizontal:__**

b Graphs of functions may be horizontally shrunk and stretched by multiplying the x-coordinate by a number. The y-coordinate will not change. b To horizontally stretch a graph, multiply the x-value of the function by a number less than 1. b To horizontally shrink a graph, multiply the x-value of the function by a number greater than 1.

Examples to do with students:

__Original function__ f(x)=(x+1)3 black __Horizontal Shrink__ (multiply x-value by 2): f(x)=(2x+1)3 red __Horizontal Stretch__ (multiply x-value by .5): f(x)=( .5x+1)3 green

Reflections Across Axes - The graph of a function may be reflected across the x-axis by multiplying the y-value of the function by -1. Example if f(x)=x2 -3x+2, then g(x)= -(x2 -3x+2) is the reflection of f(x) across the x-axis. -The graph of a function may be reflected across the y-axis by multiplying the x-value of the function by -1.
 * Reflections:**

Example: if f(x)=x2 -3x+2, then g(x)=(-x)2 -3(-x)+2 is the reflection of f(x) across the y-axis.

- Notice the x-values of the original graph stay the same on a reflection across the x-axis and y-values become their opposites - Notice the y-values of the original graph stay the same on a reflection across the y-axis and x-values become the opposite Examples: __Original function__ f(x)=x2 -3x+2 black -Reflection of f(x) across the x-axis : g(x)= -(x2 -3x+2) - Reflection of f(x) across the y-axis: g(x)= (-x)2 -3(-x)+2
 * __Reflections across x- and y-axes__**

Reflection across x axis: F(x)= -( mx+b) Reflection across y axis: F(x)= m( (-1Xx)+b)
 * In general:**

-When students are handed out their Transformation "Cheat Sheet Tables [|transformationcheatsheet.doc], the answers should be as follows: [|transformationcheatsheetanswers.doc]

-Students should have these answers for the Activity Options Sheet [|transformationactivity.doc] : [|transformationactivityanswers.doc]

[|Getting the idea: An introduction to horizontal stretching and shrinking] : This website introduces the reader to the idea of horizontal and vertical movement as well as stretching and shrinking (as well as provides pictures). This website is more of a supplemental website if a student needs another resource to look at information, because there are better websites that explain these ideas much more thoroughly. This website is consistent with using parabolas though, and this may be good for one to consider. [|Horizontal Stretch and Shrinking of Parabolas] : This site works solely with parabolas to ensure that students get the concept of horizontal and vertical transformations. The quiz at the end provides them with a quick and easy self-assessment that allows them to see if they need to practice their skills further. [|Transformations of graphs walk-through] : This site provides an excellent slide show that walks through what a transformation is, what kinds there are, how they are used and includes examples. I plan to use this website for much of my lesson. [|Vertical and Horizontal Stretches of Trig Graphs] : This website give students another medium of uniformity to think about horizontal and vertical transformations in. [|All-inclusive website of graph transformations] : A great website that provides visuals as well as a verbal explanation of each sort of graph transformation that is included in my lesson. [|Graph transformations and reflections] A very generalized explanation of graph transformations. Not very student friendly if they are not math-lovers, but is sufficient for content notes. [|Horizontal and Vertical Graph Transformation Videos] : This website is a good way for students to gain extra perspective on vertical and horizontal graph transformations. It may also make a good model for their podcasts (if they just listen and not watch) to see what parts need to be explained more specifically if they do not have a visual. [|Transformation Practice Problems] : This website provides an explantation for each of the graph transformation as well as provides great practice problems to have students identify what transformations are occurring. [|Interactive Transformation Practice] : If you scroll down the page to "Graph Transformations", there are several links that help students interact and visualize what each transformation does to the graph.
 * These are the approved websites that they can use**:

- When talking about [|Garage Band], remind students how to record, overlap sound, export and turn the metronome off. This should be enough of a refresher for how much we've used it in this lesson. · [|KWL graphic organizer] · Graph Transformation Worksheet (Activity Option worksheet) [|transformationactivity.doc] · Transformations "Cheat Sheet" Table [|transformationcheatsheet.doc] · Math Man or Graph Girl Podcast Descriptor Handout [|podcastexplanation.doc] · Podcast Rubric [|podcastrubric.xls]
 * __Handouts__**