Evidence of Student Learning - Fall 2013 by Ben Cohen

Evidence of Student Learning Benjamin Cohen Towson University EDUC 797 Internship I with Seminar Trigonometry/Pre-Calculus Grade 12 Winters Mill High School October 2013

Table of Contents InTASC and COE Rationales .................................................................................................... 4-25

Contextual Factors Contextual Factors ................................................................................................................... 28-32 References and Credits ..................................................................................................................33

Learning Objectives Learning Objectives ................................................................................................................. 36-37

Assessment Plan Assessment Alignment Summary ..................................................................................................41 Pretest....................................................................................................................................... 42-43 Post-Test .................................................................................................................................. 44-45 CCPS Summative Assessment................................................................................................. 46-49 Formative Assessments............................................................................................................ 50-67 Identification of Subgroups............................................................................................................68

Lesson Plans Pretest Item Analysis (Whole Class) ....................................................................................... 70-71 Pretest Item Analysis (Female Students) ................................................................................. 72-73 Pretest Item Analysis (Below 80% on â&#x20AC;&#x153;Speed Testâ&#x20AC;?) .............................................................. 74-75 Lesson Plan 1 ........................................................................................................................... 76-82 Lesson Plan 2 ........................................................................................................................... 88-94 Lesson Plan 3 ......................................................................................................................... 98-105 Classroom Management Matrix ...................................................................................................106 Lesson Plan 4 ....................................................................................................................... 107-114

Lesson Plan 5 ....................................................................................................................... 115-122

Instructional Decision-Making Instructional Decision-Making ....................................................................................................125

Analysis of Student Learning Assessment Data Collection (Whole Class) ................................................................................128 Pre-/Post-Test Item Analysis (Whole Class) ....................................................................... 129-131 Disaggregated Assessment Data Collection (Female Students) .......................................... 132-133 Pre-/Post-Test Item Analysis (Female Students) ................................................................. 134-136 Disaggregated Assessment Data Collection (Below 80% on “Speed Test”) ....................... 138-139 Pre-/Post-Test Item Analysis (Below 80% on “Speed Test”) .............................................. 140-142

Evaluation and Reflection Evaluation and Reflection .................................................................................................... 144-149

InTASC 1: Learner Development The teacher understands how learners grow and develop, recognizing that patterns of learning and development vary individually within and across the cognitive, linguistic, social, emotional, and physical areas, and designs and implements developmentally appropriate and challenging learning experiences. Rationale This standard is a more formal repetition of the oft-stated adage that “no two students are alike.” Rather than assume instruction is a “one-size-fits-all” endeavor—even in an advanced class such as Trigonometry/Pre-Calculus—effective teachers will realize they are still working with a group of students with differing prerequisite knowledge, abilities, and learning styles and will modify instruction accordingly. In addition, these modifications should be applied to all aspects of the lessons, not just the teacher-led portions, including the warm-up and formative assessments. For this standard, my artifact is the pretest item analysis for both the full class and two disaggregated subgroups from my first ITE in October. Although I worked with an advanced class (Trigonometry/Pre-Calculus), there is nevertheless a wide range of abilities in the class; some students had attained a near-perfect score on every assessment to that point, while others were still struggling to memorize the sine and cosine values of specified angles, a knowledge base that is crucial to success in the class. In addition, I chose to break down students’ scores by gender given the well documented national achievement gap between male and female students. When planning the unit, I kept in the back of my mind that it would likely become necessary to modify instruction to accommodate these two subgroups, but I also wanted to get a sense of where these students stood relative to their peers at the start of the unit.

This artifact fits the “understanding the context for learning” portion of CPTAAR by showing my understanding of students’ diverse learning styles and needs, as well as recognizing that it is imperative to address these in an appropriate manner when planning both a full unit and individual lessons. As I stated in the initial paragraph, teaching is not a “one size fits all” endeavor, nor is there a single best way to teach. Proper instruction takes into account students’ prerequisite knowledge, skills and abilities, diverse needs, and the contextual factors of both the school and the local environment; the pretest item analysis is just one pillar on which this ideal instructional planning and understanding rests. This artifact can be found on pages 70-75 of the ESL.

InTASC 2: Learning Differences The teacher uses understanding of individual differences and diverse cultures and communities to ensure inclusive learning environments that enable each learner to meet high standards. Rationale This standard emphasizes that an effective instructor must take into account the contextual factors of the school and community in which (s)he is teaching to ensure that all learners can relate to the lesson. For example, a math teacher in an area with many families on the lower end of the socioeconomic spectrum should not ask students to solve geometry problems involving the circumference of a pool in the backyard of a house; many students would likely not be able to relate to this and would become disconnected from the lesson. Similarly, a history teacher in a heavily Jewish area would do well to avoid planning a December lesson on the history of Christmas celebrations in the U.S. For this standard, I have chosen the second lesson plan from ITE week. This lesson included a number of applications of trigonometric functions, so I felt it was particularly important to keep in mind the high percentage of FARMS students at my high school when planning it. All examples used in this lesson were carefully selected to ensure they represented experiences that all students, regardless of socioeconomic status, would have had the chance to experience, such as attending a school volleyball match. The goal was to give all students examples to which they could easily relate, thus allowing them to fully realize the power of trigonometric functions and achieve to the best of their abilities, rather than unfamiliar examples that would have only served to make them feel disconnected from my instruction. This artifact fits into the “understanding context for learning” component of the CPTAAR cycle. The artifact—and the entire standard—focuses on planning instruction that is

appropriate given the demographics of the area, and in order to do this, it is essential to understand the contextual factors of the school. Without this knowledge and understanding, any time spent on planning may simply turn out to be wasted effort, for if a lesson is not planned with the students in mind, it will become significantly more difficult to meet the learning objectives for that day. This artifact can be found on pages 88-94 of the ESL.

InTASC 3: Learning Environments The teacher works with others to create environments that support individual and collaborative learning, and that encourage positive social interaction, active engagement in learning and selfmotivation.

Rationale Standard 3 emphasizes the importance of creating a classroom environment in which all students will push themselves to achieve to their fullest potential, take an active interest in their own learning, and support each other throughout the learning process. Students should not feel anxious or intimidated in any classroom, nor should they dread attending a particular class. Rather, they should look forward to spending time in a classroom whose environment helps them grow both as students and as people. My artifact for this standard is the lesson plan from my third ITE lesson, which also includes a classroom management matrix. As has been stated in previous rationales, I worked with an advanced class (Trigonometry/Pre-Calculus), but working with high-ability students does not simply mean I can walk into the classroom and expect the lesson to teach itself. There is one student in the class with a 504 plan, but above all, it is key to remember I am still working with 17-year old high school students. If I fail to keep them busy and actively engaged, they will likely venture off topic; as such, I must be careful to minimize the â&#x20AC;&#x153;down timeâ&#x20AC;? in class, which provides a fertile breeding ground for misbehavior. In addition, it is imperative that I have all of my instructional materials ready at the start of class each day, rather than rushing to the academy planning room to make copies just before the bell. Lastly, I must not underestimate the effect of my physical presence on studentsâ&#x20AC;&#x2122; behavior; they are far more likely to remain on task if I am

moving throughout the classroom instead of standing in the front for 80 minutes. The classroom management matrix represents the codification of these ideas. This artifact helps represent the â&#x20AC;&#x153;planning instructionâ&#x20AC;? component of the CPTAAR cycle. Classroom management considerations are arguably the most crucial part of planning any lesson, and it is crucial to devote as much time and effort to them as to the instructional procedures. A well-planned class includes not only an active, engaging instructional component, but also a series of classroom management strategies that work to ensure all students are involved in the lesson and everyone leaves the room having gotten more out of the lesson than simply another page of notes. The classroom management matrix can be found on page 106 of the ESL; the accompanying lesson plan can be found on pages 98-105.

InTASC 4: Content Knowledge The teacher understands the central concepts, tools of inquiry, and structures of the discipline(s) he or she teaches and creates learning experiences that make these aspects of the discipline accessible and meaningful for learners to assure mastery of the content.

Rationale This standard represents a two-pronged approach to effective instruction: In addition to having a thorough command of his/her subject area (including the “why” and not just the “what”), a teacher must be able to translate that knowledge into terms that students can understand. To use a math example, it is not sufficient to know that the product of two negative numbers is positive, nor would it be acceptable to tell students that such a product is positive because it can be written as the multiplicative inverse of another integer’s multiplicative inverse. Rather, I must be able to put it into language appropriate for the students’ knowledge base and level of study: Making a negative number negative is tantamount to taking the opposite of its opposite; in much the same way the opposite of the opposite of up (i.e., the opposite of down) is up, so too is the negative of a negative number positive. For this standard, I chose as my artifact the lesson plan from the first day of my ITE. This lesson began the class’s unit on trigonometric functions, which I have long believed to be the most difficult unit in a standard pre-calculus class. As such, I attempted to maximize the extent to which students would find this material accessible and meaningful; for example, rather than simply devote the lesson to an introduction to periodic functions, I included several realworld examples to which students could relate, such as modeling sound waves and tides. This lesson even included a comparison of the sound waves of three guitar solos in an attempt to engage all learners. It would not have been possible to do this without an in-depth knowledge of

the topic and its seemingly endless real-world applications; at the same time, I had to make sure to put this information into a form that students would be able to grasp. This artifact aligns with the “planning instruction” and “teaching” components of the CPTAAR cycle. When planning a lesson, it is key to ensure that all of the material in it will be accessible to students, both by being on a level they can understand and by being interesting and engaging. However, this is only half of the job; it is also necessary to teach it in such a manner and be prepared to adjust “on the fly” and express it in a different form if necessary to improve students’ comprehension. The artifact can be found on pages 76-82 of the ESL.

InTASC 5: Application of Content The teacher understands how to connect concepts and use differing perspectives to engage learners in critical thinking, creativity, and collaborative problem solving related to authentic local and global issues.

Rationale This standard states that teachers should be able to effectively connect concepts taught in a given class to both students’ work across the curriculum and their experiences outside of school. For example, students studying the solar system and planetary motion in science class could have their knowledge augmented by a discussion of the controversy surrounding the heliocentric solar system models put forth by Nicolaus Copernicus and Johannes Kepler, rather than treating their science and history classes as two disparate entities. To provide another example, a mathematics unit on irrational numbers could also ask students to consider why they caused such an uproar in the religious societies in which they were first discovered to help sharpen their critical-thinking skills. My artifact for this standard is a PowerPoint used on the first day of my ITE. This PowerPoint includes a series of real-world applications of trigonometric functions—sound waves from a guitar, modeling ocean tides, and examining one’s blood pressure. It has long been my feeling that much of students’ ambivalence towards mathematics is a direct consequence of the tactic of teaching it in a vacuum, as well as inadequate responses to the oft-heard question, “When am I ever going to use this?” As such, I felt it was crucial to expose students to the numerous applications of trigonometric functions not only throughout the unit, but also on the first day of the unit as a “hook.” I also wanted to use problems to which students can legitimately relate in their everyday lives, rather than, say, cell tower triangulation.

This artifact fits the “understanding the context for learning” portion of the CPTAAR cycle; one can also argue that it fits the “planning instruction” component of the cycle. As was stressed in the previous paragraph, it is not enough to merely include a real-world or crosscurricular connection for the sake of it; students must truly be able to understand and relate to it. In addition, I had to take particular care when planning cross-curricular applications for this unit due to the high percentage of FARMS students at the school (nearly 25 percent of the student population). This artifact can be found on pages 83-86 of the ESL. The corresponding lesson plan can also be found on pages 76-82.

InTASC 6: Assessment The teacher understands and uses multiple methods of assessment to engage learners in their own growth, to monitor learner progress, and to guide the teacher’s and learners’ decision making.

Rationale This standard emphasizes the importance of using multiple methods of assessment beyond simply homework, quizzes, and tests. These assessments can take many forms, from informally observing students as they are working individually to assigning a special essay or project at the end of a unit as the summative assessment. In addition, as the standard notes, it is key to ensure these assessments are being used for the proper purposes—not just to give students a taste of something different, but to help the students grow and realize their full potential and provide an additional means of assessing their knowledge. For this standard, I chose as my artifact the lesson plan from the fourth day of my ITE. This lesson included various assessment techniques, beginning with the warm-up, which was designed to gauge students’ progress to that point. For the warm-up, I asked a series of five true/false questions, asking students to hold up notecards reading “YES” or “NO” to answer each; their answers guided both the remainder of that lesson as well as my planning for the subsequent one. In addition, the formative assessments for that day contained a mixture of problems; in addition to graphs, students were given a series of equations for trigonometric functions and asked to identify the key components of each. Also, like with all of the ITE week lessons, I used observations of students as an informal assessment technique as well. This artifact aligns with the “planning instruction” and “assessing student learning” components of the CPTAAR cycle. Clearly, the standard focuses on assessment, and I have

chosen to highlight the various assessment techniques detailed in the artifact. However, merely using multiple methods of assessment is insufficient. It is also crucial, as the standard explains, to allow the results of these assessments to guide the planning process; that was certainly the idea behind my assessments, and this strategy can and does lead to highly effective instruction. The artifact can be found on pages 107-114 of the ESL. Studentsâ&#x20AC;&#x2122; aggregated responses to the true/false questions are electronically highlighted and can be found on page 112.

InTASC 7: Planning for Instruction The teacher plans instruction that supports every student in meeting rigorous learning goals by drawing upon knowledge of content areas, curriculum, cross-disciplinary skills, and pedagogy, as well as knowledge of learners and the community context.

Rationale This standard, which is similar to InTASC 5, states that effective instructors should demonstrate both thorough knowledge of their content areas and pedagogy and the ability to integrate cross-curricular connections to which students can relate. Content should not be taught in a vacuum, nor should lessons include connections that, while technically accurate, students cannot be reasonably expected to understand or relate to. In addition, knowledge of the community is essential when planning lessons—for example, a geometry class in a New York City school should not include problems asking students to find the acreage of the front yard of a house, since this is not something to which most New York residents can relate. My artifact for this standard is a PowerPoint from the second day of my ITE. Much as I did during the first lesson, I made it a priority in this lesson to integrate examples to which students could legitimately be expected to relate—in this case, the bell and fire alarm in the school building. In this case, the idea was to introduce students to the concepts of frequency and period changes by comparing and contrasting a dull, low-pitched sound to a shriller, higherpitched one. Although the possibilities for examples are nearly limitless, as a nod to the high percentage of FARMS students at the school I chose to use examples that would be indisputably familiar to every student at the school, rather than, say, a ship’s foghorn and train whistle. This artifact reflects the “understanding the context for learning” and “planning instruction” components of the CPTAAR cycle. As was stated in the previous paragraph,

understanding the community context is essential when planning cross-curricular connections to integrate into lessons; in addition, these must be effectively woven into the instructional plan for the day. Any connections to which students cannot relate represent merely wasted planning effort; in addition, their utility will be significantly limited if they seem like a last-minute throwin to the lesson rather than a seamlessly integrated component of it. This artifact can be found on pages 95-97 of the ESL.

InTASC 8: Instructional Strategies The teacher understands and uses a variety of instructional strategies to encourage learners to develop deep understanding of content areas and their connections, and to build skills to apply knowledge in meaningful ways.

Rationale This standard emphasizes that effective instruction can and should take multiple forms. Particularly in mathematics, there has been a recent movement away from the teacher-directed lesson towards a more student-centered model; however, this standard states that math class should also not simply consist of doing discovery activities every day, but should have an effective mix of student- and teacher-led components. In addition, lessons should focus on acquiring both sufficient breadth and depth of content knowledge, rather than being merely an overview of a series of topics. My chosen artifact for this standard is the lesson plan from the first day of my ITE. Although this lesson necessarily included a heavier lecture component than the subsequent classes, since it served as the introduction to the unit, I attempted to employ a number of different instructional strategies to ensure students remained engaged throughout the lesson. The PowerPoint, which may have appeared on the surface to be merely a tool to “spice up” the lecture, in fact included several opportunities for students to think deeply about the material being presented. When planning the lesson, I included elements designed to appeal to all types of learners; the examples presented in the PowerPoint were not intended to be a crutch for the lecture, but rather a means for highlighting these elements. This artifact closely parallels the “planning instruction” and “teaching” portions of the CPTAAR cycle. Certainly, any artifact that focuses on instructional strategies aligns with the

“teaching” portion of CPTAAR by definition, and this one is no exception; the use of differentiated instruction is a two-step process that includes both planning and delivery. However, it is also extremely difficult, if not impossible, to simply “wing it” during the lesson; instead, the techniques of differentiated instruction should be well thought out and planned in advance of the lesson. In order to maximize the gains from various instructional strategies, adequate time and effort must be allotted to both of these phases. This artifact can be found on pages 76-82 of the ESL.

InTASC 9: Professional Learning and Ethical Practice The teacher engages in ongoing professional learning and uses evidence to continually evaluate his/her practice, particularly the effects of his/her choices and actions on others (learners, families, other professionals, and the community), and adapts practice to meet the needs of each learner. Rationale This standard emphasizes the need for constant reflection and self-evaluation in the teaching field. It has long been my feeling that teaching is not a one-way instructional street; while the primary objective is for students to learn from their teachers, teachers can learn just as much from their students. Effective instructors will make every effort to take advantage of workshops, conventions, webinars, and other professional-development opportunities that present themselves. In addition, this serves as another reminder that teaching is not a “one size fits all” pursuit; each student has his/her own unique needs that must be taken into account when planning instruction. My artifact for this standard consists of two items: feedback received from my university supervisor from the first lesson during ITE week, and the following day’s lesson plan, showing implementation of that feedback. Although the suggestion may appear to be minor—handing out classwork at the start of class rather than in the middle of it—it is crucial to note that this feedback represented the first formal assessment of my instruction, and I felt it was imperative to establish the precedent that I am constantly seeking and implementing feedback from more experienced professionals in the field. Merely dismissing this suggestion as “inconsequential” or “unimportant” would have made me appear to be a know-it-all, which I feel is certainly not the case for someone who has been teaching for several decades, let alone two

months. Rather, I felt it was crucial to make it clear from the outset that I will constantly strive to reflect on and improve my teaching strategies and am willing to consider and integrate any piece of advice, no matter how small. This artifact fits into multiple components of the CPTAAR cycle: planning instruction, teaching, and reflecting. The first two are self-evident: I took this feedback into account when planning and delivering subsequent lessons. In addition, I considered the feedback I received when reflecting on the lesson after school that day, something I now do following each lesson I teachâ&#x20AC;&#x201D;a self-examination of what went well, what I could have done differently, and the steps I must take to continually improve, beginning with the next dayâ&#x20AC;&#x2122;s lesson. The feedback from my university supervisor can be found on page 87. The lesson plan showing implementation of this feedback can be found on pages 88-94; the relevant section, on pages 90-91, is electronically highlighted as well.

InTASC 10: Leadership and Collaboration The teacher seeks appropriate leadership roles and opportunities to take responsibility for student learning, to collaborate with learners, families, colleagues, other school professionals, and community members to ensure learner growth, and to advance the profession.

Rationale This standard focuses on teachersâ&#x20AC;&#x2122; professional growth. Certainly, it should be the goal of every teacher to see his/her students achieve to the best of their abilities, and seeking leadership roles and opportunities represents progress towards that goal. For example, during the first few years of oneâ&#x20AC;&#x2122;s teaching career, a new teacher would likely rely on colleagues and his/her department chair for advice on pedagogical techniques. After several years, however, that teacher will likely have acquired a knowledge base that will allow him/her to move up to the position of department chair or a similar position from which (s)he can in turn mentor other new teachers, perpetuating the cycle of collaboration and peer support. For this standard, I have chosen my ITE week evaluation and reflection. As a student teacher, my opportunities to assume a leadership role within the department, school, and community are understandably limited, but I am still eagerly seeking opportunities to allow me to grow professionally and, whenever possible, both receive and give pedagogical knowledge. I discussed many of the points raised in this evaluation and reflection in detail with my mentor following ITE week, and I have also attempted to build connections with additional members of the math department to increase the knowledge pool from which I can borrow. I feel it is critically important to make sure I am constantly doing everything in my power to improve studentsâ&#x20AC;&#x2122; learning outcomes, and this is just one step in that ongoing process.

This artifact mirrors the “analysis” and “reflection” components of the CPTAAR cycle. Certainly, these represent two actions that should be taken by any teacher following every lesson, but as this represented my first experience planning and delivering instruction, I felt it was particularly important in my case to engage in this process. In addition to a period of selfreflection following each lesson, I took time at the conclusion of ITE week to look back and reflect on the positives and negatives of my five days of instruction. This artifact can be found on pages 144-149 of the ESL.

COE 11: Use of Technology The teacher views technology not as an end in itself, but as a tool for learning and communication, integrating its use in all facets of professional practice, and for adapting instruction to meet the needs of each learner.

Rationale This standard focuses on one of the most crucial aspects of 21st-century instruction, technology. As the standard attempts to make clear, technology for technology’s sake is not sufficient to improve a lesson; rather, there must be a clear purpose to the implementation of technology. For example, rather than simply using Geometer’s Sketchpad to demonstrate how to graph a limaçon and rose curve, an effective lesson would use Geometer’s Sketchpad to illustrate why a limaçon and rose curve have their distinctive shapes. Similarly, merely using a technologically advanced calculator such as a TI Nspire does not improve instruction; it is key to determine how to effectively integrate the Nspire into the lesson and make full use of its capabilities. My chosen artifact for this standard is the lesson plan from the fifth day of my ITE. To be fair, I attempted to include a significant technological component in all five lessons during the ITE. However, I had attended a TI Nspire workshop over the weekend and was eager to begin implementing the applications and tips I had learned when planning this lesson. For example, I loaded the graphs of all functions that I would be demonstrating onto students’ calculators prior to class; the idea behind this was to ensure students were focusing on transformations of the functions, rather than simply their graphs. In addition, I devoted part of this lesson to tips for graphing the tangent and cotangent functions on a TI-83 and TI-84, but made sure to frame this topic in a manner that would not unnecessarily exclude students who do not have these items.

This artifact closely parallels the “planning instruction” and “teaching” portions of the CPTAAR cycle. Once again, it is crucial to both properly plan a lesson that will include a technology component to ensure it isn’t simply technology for technology’s sake and also to teach it in a manner that will ensure maximum utilization of the available technology. In addition, it can be argued that this aligns with the “understanding the context for learning” component; as was stated above, during the discussion of how to graph these functions on a TI-83 and TI-84, I made sure to frame it in a manner that would not exclude low-SES students (e.g., I did not ask which students had a TI-83 or TI-84, but simply introduced this as a general tip for these calculators). This artifact can be found on pages 115-122 of the ESL.

Section 1: Contextual Factors

Contextual Factors Winters Mill High School (WMHS), located in Westminster, Maryland, is one of the newest additions to the Carroll County Public Schools system, having opened in 2002. Carroll County is the ninth-most populous of Marylandâ&#x20AC;&#x2122;s 24 counties (including Baltimore city), with a population of 167,134, according to the 2010 U.S. census, but is the sixth-least populous when the sparsely populated Eastern Shore counties (all of which have lower populations than Carroll County) are excluded. The county is also predominantly white; according to the 2010 census, 92.9 percent of the population is white. This is reflected in the school demographics: of the 1,108 students at WMHS, 955 (86.2 percent) are white. An additional 103 students (9.2 percent) are African-American. The school is split nearly evenly by gender, with 542 female students and 566 males. Carroll County is also exceptionally wealthy, with a median family income of $90,376, according to a 2007 estimate by the census bureau. While one would expect this to lead to a high level of education spending, the county has the second-lowest level of spending per student-$12,230â&#x20AC;&#x201D;among the suburban Baltimore and Washington counties, ahead of only Harford County. In addition, the high median income in the county does not mean that every student in the school is from a high-socioeconomic background: Currently, 258 students (23.3 percent) are eligible for free and reduced-price meals, the highest percentage in Carroll County. In general, classrooms at WMHS are well equipped with regard to technology. All classrooms have projectors and desktop computers; desktop computers are equipped with ActivInspire software, which allows the instructor to annotate a projected image with the use of a handheld device. WMHS classrooms are not equipped with SMART Boards, however, which limits the extent to which interactive computer media can be utilized. This classroom does have a coordinate plane whiteboard at the front of the room, however. The classroom also includes 30 TI Nspire calculators as part of a pilot project in the school. Each day at WMHS features four

80-minute periods in addition to a 25-minute lunch period and 35-minute advisory period; the class I will be teaching, Trigonometry/Pre-Calculus, meets daily during mod 4, which meets from 1:00-2:20 p.m. There are 16 students enrolled in Trigonometry/Pre-Calculus; enrollment is split almost evenly on a gender basis, with nine female students and seven males. At the beginning of the semester, students were asked to fill out surveys that asked, among other details, about students’ learning styles; many students indicated a preference for interactive activities, and a few also noted that there are certain individuals in the class with whom they do not wish to be paired for group activities. There is one student in the class with a 504 plan; due to a hearing impairment, he must be seated at the front of the classroom during lessons. (There are also seven students with “medical alerts,” but these pertain to conditions that do not affect their learning outcomes, according to the instructor.) There is only one minority student in the class, an African-American male. All members of the class are 12th graders who have previously taken Intermediate Algebra, Geometry, and Algebra II, in that order; during the previous academic year, five students received grades of “A” in Algebra II, seven received a “B,” and four received a “C.” This is a Level 6 (“Academic”) class, which represents a standard-level curriculum.1 It is expected that the following school factors will influence my instructional decisions: •

High percentage of FARMS students. During one lesson I observed during the previous

school year at another Carroll County school, students worked during class to solve geometry problems involving the amount of fencing required to fully enclose a home’s backyard and the circumference and area of an above-ground pool in the backyard. At wealthier schools, it is possible that these items can be taken for granted, but at a school like WMHS, where nearly onequarter of the student body is eligible for FARMS—again, the highest percentage in Carroll !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 1

There are three levels of classes at WMHS: Level 1 (foundational), Level 6 (academic), and Level 8 (honors/AP).

County—great care must be taken to ensure these students are not made to feel uncomfortable or given problems involving situations to which they cannot relate. For example, instead of asking for the amount of fencing required to enclose a yard, a problem could ask for the amount of paint needed to paint a horizontal line on the classroom walls. Certainly, one can argue that such problems should be generic and unambiguously applicable to all students regardless of the demographics of a school; regardless, this takes on added importance at a school with a high percentage of low-SES students, such as WMHS. •

Availability of technology. While the lack of a SMART Board limits opportunities for

interactive activities—such as using a graphing calculator emulator to check the solution to questions in front of the whole class—the availability of Nspires presents significant opportunities for the integration of technology. The calculators can be equipped with wireless devices that will allow me to send and receive documents to and from all students’ calculators— this could be used as a formative assessment technique, for example. In addition, the ActivInspire device mentioned earlier allows users to annotate the screen wirelessly; although the use of this device is restricted to instructors, it will allow me to circulate the room and ask questions—such as what the graph of a given function will look like—instead of simply remaining in the front of the room at the computer to do so. (It should also be noted that the students are not yet expected to be proficient in the use of the Nspire calculators, since they were obtained as part of a pilot project. Thus, part of the instruction in this unit must also be devoted to calculator use, as it has been throughout the semester in this class.) In addition, the following student characteristics should be kept “front and center” as I plan: •

Gender and Racial Differences. One well documented national phenomenon is the

dearth of women in mathematics-based positions (e.g., college professorships), as well as an

achievement gap between female students and their male peers. A long-standing personal observation is that word problems in math classes tend to be male-centric; for example, sports analogies most frequently involve football or baseball, rather than gender-neutral sports such as golf or tennis. To combat this, I plan to use word problems and real-world applications that apply to both genders, such as using trigonometric functions to compare the velocities of Sally Ride and John Glennâ&#x20AC;&#x2122;s respective space shuttles. Although a minor step, research has suggested this may help get female students more involved in their math classes. In addition, it has also been shown that the introduction of technology can virtually eliminate the achievement gap between male and female students (Dixon, Cassady, Cross, & Williams, 2005). Similarly, I must utilize culturally responsive teaching techniques to avoid ostracizing the one African-American student in the class. As with male and female students, there is a long-standing achievement gap in mathematics between white students and their minority counterparts, with some research suggesting that this is due in part to a disconnect similar to the one felt by female students (Ford & Whiting, 2008). Again, it is crucial to ensure my efforts to ensure the needs of one student are met do not result in suboptimal outcomes for the rest of the class; I plan to accomplish this primarily through integrating references to the contributions of both white (e.g., Euclid) and African-American mathematicians (such as Benjamin Banneker, a former slave from modernday Baltimore County) into my lessons. â&#x20AC;˘

504 Plans. There is only one student with a 504 plan in the class; as was mentioned

earlier, he has a hearing impairment and so must be seated in the front of the classroom. I also feel it would be advantageous to help him develop a nonverbal signal to me if he cannot hear what I am saying, such as if I am in the back of the classroom utilizing the ActivInspire device,

rather than having to raise his hand and inform he that he cannot hear. This ensures his needs are met while helping him avoid embarrassment. As was mentioned earlier, all students in this class have previously taken Algebra I, Geometry, and Algebra II, in that order, according to the CCPS curriculum guide. Students will be expected to be familiar with symmetry and transformations of functions (horizontal and vertical stretches, compressions, and shifts), which were covered in Algebra II, as well as sine, cosine, and tangent values for specified angles, which were covered in an earlier unit. This unit is an expansion of their knowledge of function symmetry and transformations; whereas they previously were only familiar with functions that were symmetric about an axis, they will now be introduced to functions whose symmetry repeats infinitely (sinusoidal functions), with the help of their knowledge of trigonometric values. An analysis of the pretest suggests that students have retained a significant amount of content knowledge from Algebra II but, as expected, are unfamiliar with periodic functions. Fifty percent of students (eight out of 16) correctly identified a vertical translation, while 25 percent properly identified a phase shift; although I will still need to introduce this as entirely new material, it is likely these topics can be covered in one day each, rather than the two days suggested by the CCPS curriculum guide. At the same time, only one student was able to graph a sine and cosine function, and none were able to graph a secant or cotangent function; while students are expected to have a thorough knowledge of trigonometric values, they have not yet had any experience translating these values into a graph. In addition, only one student was able to solve a problem involving simple harmonic motion, and none was able to determine a model for such a problem. Instruction on these topics will make use of studentsâ&#x20AC;&#x2122; existing knowledge of trigonometric values; it is also clear significant time will need to be devoted to them. Applications will also be integrated in the lessons so students are not learning in a vacuum.

References and Credits Dixon, F., Cassady, J., Cross, T., & Williams, D. (2005). Effects of technology on critical thinking and essay writing among gifted adolescents. The Journal of Gifted Secondary Education, 16, 180-189. Ford, D. Y., Grantham, T. C., & Whiting, G. W. (2008). Another look at the achievement gap: Learning from the experiences of gifted black students. Urban Education, 43, 216-239. doi:10.1177/0042085907312344

Section 2: Learning Objectives

Learning Objectives Grade: 12 Class: Trigonometry/Pre-Calculus Unit: Graphing Trigonometric Functions Lesson 1 Topic: Sine and Cosine Functions Curriculum Objective: Graph the sine and cosine functions, including amplitude and period, as well as vertical translations of these functions. Maryland Content Standards: 2.1.2 The student will identify the domain, range, the rule or other essential characteristic of a function. 2.2.3 The student will perform translations, reflections, and dilations on functions. Student Objective: Students will be able to graph the sine and cosine functions, including vertical translations, and will be able to identify the amplitude and period of these functions. They will also be able to identify the translations in either a given function or its graph. Lesson 2 Topic: Horizontal Translations Curriculum Objective: Graph horizontal, vertical, and combinations of translations of basic sine and cosine functions. (Note: This objective is used for consecutive lessons in the curriculum guide.) Maryland Content Standards: 2.1.2 The student will identify the domain, range, the rule or other essential characteristic of a function. 2.2.3 The student will perform translations, reflections, and dilations on functions. Student Objective: Students will be able to graph horizontal translations of the sine and cosine functions and will be able to identify the period and phase shift of these functions. Lesson 3 Topic: Combinations of Translations Curriculum Objective: Graph horizontal, vertical, and combinations of translations of basic sine and cosine functions.

Maryland Content Standards: 2.1.2 The student will identify the domain, range, the rule or other essential characteristic of a function. 2.2.3 The student will perform translations, reflections, and dilations on functions. Student Objective: Students will be able to graph combinations of translations of the sine and cosine functions and will be able to identify their amplitude, vertical shift, period, and phase shift. Lesson 4 Topic: Cosecant and Secant Functions Curriculum Objective: Graph the secant and cosecant functions, as well as horizontal, vertical, and combinations of translations of these functions. Maryland Content Standards: 2.2.3 The student will perform translations, reflections, and dilations on functions. Student Objective: Students will be able to graph the cosecant and secant functions, including horizontal and vertical translations and combinations of these translations. Lesson 5 Topic: Tangent and Cotangent Functions; Simple Harmonic Motion Curriculum Objectives: Graph the tangent and cotangent functions, as well as horizontal, vertical, and combinations of translations of these functions. Solve problems involving simple harmonic motion. Maryland Content Standards: 2.1.2 The student will identify the domain, range, the rule or other essential characteristic of a function. 2.2.3 The student will perform translations, reflections, and dilations on functions. Student Objective: Students will be able to graph the tangent and cotangent functions, including horizontal and vertical translations and combinations of these translations. They will also be able to use the trigonometric functions they have learned to solve problems involving simple harmonic motion.

Section 3: Assessment Plan

Assessment Alignment Summary The following chart provides a summary of the assessment plan for the Graphing Trigonometric Functions unit, illustrating the relationships between items on the pre-test, post-test, formative assessments, and county-provided summative assessment (not used in this unit). (The assessments listed herein exclude informal assessment techniques such as instructor observations.) For each lesson, the relevant objective(s) are listed as well. In some cases, objectives apply to multiple formative assessments due to the cumulative nature of this unit; in these cases, the first lesson number listed is when the objective was formally introduced, with subsequent lessons making use of the same topic but having a different formal objectiveâ&#x20AC;&#x201D;for example, students were introduced to vertical translations in lesson one; this was also assessed in lesson three, which concerned combinations of translations. Note: For brevity, the Maryland Content Standards for this unit are listed only by their numbers in the following chart. The full standards are as follows: 2.1.2 The student will identify the domain, range, the rule or other essential characteristic of a function. 2.2.3 The student will perform translations, reflections, and dilations on functions.

Trigonometry/Pre-Calculus Graphing Trigonometric Functions – Pre-Test

Name: ______________________ October 8, 2013

Part I. Choose the one alternative that best completes the statement or answers the question. 1 π 1. Find the vertical translation of y = 2 + sin( x − ) 2 2

A) Down

π 2

B) Up

1 2

C) Up 2

D) Down 2

2. Find the phase shift of the graph of y = 2 tan(6x + π ) . A) π units to the left

B) 2π units to the left

C) π units to the right

3. Find the period of y = −3csc(2x + A) 2

π ). 4 B) 3

C) π

4. Find the amplitude of y = − sin(3x + π ) . A) 3

π units to the left 6

B) 1

π 3

π 4

D) π

Part II. Graph the following functions over the specified interval. 5. y =

1 cot(x) (one period) 4

6. y = sec(2x) + 1 (two periods)

7. y = −2sin(x + π ) (two periods)

8. y =

1 π cos(x − ) (two periods) 2 2

Part III. Solve the problem. 9. The position of a weight attached to a spring is given by s(t) = −5 cos(3π t) , where t represents the time, in seconds, and s(t) represents the position, in inches, at time t. When does the weight first reach its maximum height?

10. A weight attached to a spring is pulled down 3 inches below the equilibrium position. Assume the 3 frequency is cycles per second. Determine a function s(t) that gives the position of the weight at π time t seconds.

Trigonometry/Pre-Calculus Graphing Trigonometric Functions – Post-Test

Name: ______________________ October 14, 2013

Part I. Choose the one alternative that best completes the statement or answers the question. 1. Find the vertical translation of y = −3 + sin(2(x − A) Down

π 4

B) Up 3

2. Find the phase shift of the graph of y = A)

π units to the left 2

π units to the right 2

π )) 4 C) Down 3

1 π tan(3(x − )) . 2 2

D) Up 2

3π units to the right 2

D) 3 units to the right

1 3. Find the period of y = − csc( (x + π )) . 2

A) 2

B) 4

4. Find the amplitude of y = 1− 2sin(x + A) 2

C) 4π

π ). 4

B) 1

π 4

π 2

Part II. Graph the following functions over the specified interval. 1 5. y = cot( x) (one period) 2

6. y = sec(x − π ) − 1 (one period)

7. y = 3sin(x −

π ) (two periods) 2

8. y = cos(2(x + π )) (two periods)

Part III. Solve the problem. 9. The position of a weight attached to a spring is given by y = −4 cos(π t) + 1 , where t represents the time, in seconds, and s(t) represents the position, in inches, at time t. When does the weight first reach its maximum height?

10. A weight attached to a spring is pulled down 8 inches below the equilibrium position. Assume the 5 frequency is cycles per second. Determine a function s(t) that gives the position of the weight at π time t seconds.

Unit%3%Exam% % % % % % % % Name___________________________________%% MULTIPLE(CHOICE.((Choose(the(one(alternative(that(best(completes(the(statement(or(answers(the(question.( Find(the(phase(shift(of(the(graph(of(the(function.( 1)%%y%=%%2%cos%(%6x%+%π)%(Aligned(with(question(2(on(preE(and(postEtests.)( % π A)%%6π%units%to%the%right%% B)%%%%%%%units%to%the%left%% 2 % π C)%%2π%units%to%the%right%% D)%%%%%%%units%to%the%left% 6 % %% % Find(the(specified(quantity.( π⎞ ⎛ y = −1 + 3sin ⎜ 4 x + ⎟ 2) Find%the%vertical%translation%of%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%.%(Aligned(with(question(1(on(preE(and(postEtests.)( 4⎠ ⎝ % π π A)%%%% Down B)%%Down%1%% C)%%% Up D)%%Up%%4% 4 4 % %% % π⎞ ⎛ 3)%%Find%the%vertical%translation%of%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%.%(Aligned(with(question(1(on(preE(and(postEtests.)% y = 5 + 2 sin ⎜ 6 x + ⎟ 6⎠ ⎝ % 1 π A)%%%% % Up B)%%Down%5%% C)%%%% Down D)%%Up%5% 6 6 % %% % π⎞ ⎛ y = −2sin ⎜ 4 x + ⎟ 4)%%%Find%the%period%of%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%.%(Aligned(with(question(3(on(preE(and(postEtests.)% 2⎠ ⎝ % π A)%%2%% B)%%4%% C)%%π%% D)%%% 2 % %% % Give(the(amplitude(or(period,(as(requested.( 5)%%%Period%of%y%=%sin%5x%(Aligned(with(question(3(on(preE(and(postEtests.)% % A)%% 2π B)%%5%% C)%%1%% D)%%2π%

% %% 6)%%Amplitude%of%y%=%-4%sin%5x%(Aligned(with(question(4(on(preE(and(postEtests.)% % A)%%π B)%%π C)%% 4 %% %%

% % ( ( ( ( ( (

D)%%4%

SHORT(ANSWER.((Write(the(word(or(phrase(that(best(completes(each(statement(or(answers(the(question.( Graph(the(function(over(a(one-period(interval.( %% 1 π⎞ ⎛ 7)%%%%% y = 1% + cos %4 ⎜ x + % ⎟ (Aligned(with(question(7(on(preE(and(postEtests.) 2 2 ⎝ ⎠ % % % % % % % % % % % ( ( Graph(the(function.(( ( 1 2 π 8)%%%%y = tan% ⎛⎜ x + % ⎞⎟ (Aligned(with(question(5(on(preE(and(postEtests.)% 2 5 2 ⎝ ⎠ % % % % % % % % % % % % % % % % 9)%%%%y = 2 cot% x (Aligned(with(question(5(on(preE(and(postEtests.)% 3 % % % %

⎛

π⎞

10)%%%%y = sec ⎜%x − ⎟ % 2⎠ ⎝ %

% % % % %% % 3 ⎛ π⎞ 11)%%%%y = sin% ⎜ x − ⎟% 4 3⎠ ⎝ %

% % %% %% % % 12)%%y%=%sin%πx% %

% % %

(Aligned(with(question(6(on(preE(and(postEtests.)%

(Aligned(with(question(7(on(preE(and(postEtests.)%

(Aligned(with(question(8(on(preE(and(postEtests.)%

13)%%%%y = cos% x 2 4 %

(Aligned(with(question(7(on(preE(and(postEtests.)%

% % %% % 14)%%y%=%cos%%2x% %

(Aligned(with(question(8(on(preE(and(postEtests.)(

% % % %% % Solve(the(problem.( 15)%%A%weight%attached%to%a%spring%is%pulled%down%5%in.%below%the%equilibrium%position.%Assuming%that%the% frequency%is%

%cycles%per%sec,%determine%a%model%that%gives%the%position%of%the%weight%at%time%t"seconds.%

(Aligned(with(question(10(on(preE(and(postEtests.)% % %% % 16)%%The%position%of%a%weight%attached%to%a%spring%is% s (t ) = −2cos 7π t "inches%after%t%seconds.%% When%does%the%weight%first%reach%its%maximum%height?% (Aligned(with(question(9(on(preE(and(postEtests.)% % % % % 17)%%The%position%of%a%weight%attached%to%a%spring%is% s (t ) = −4cos 7t inches%after%t%seconds.%% What%are%the%frequency%and%period?% (Aligned(with(question(9(on(preE(and(postEtests.)% % %

Trigonometry/Pre-Calculus Graphing Sine and Cosine Functions

Name: ______________________ October 8, 2013

Part I. Graph each function over a two-period interval and give its amplitude. Be sure to label the axes! 1. y = 3sin(x)

2. y = 2 cos(x)

3. y = cos(x) â&#x2C6;&#x2019; 2

4. y = sin(x) + 3

5. y = 4 sin(x) â&#x2C6;&#x2019; 1

6. y = cos(x) + 1

Part II. Identify the functions graphed below. Each graph has been transformed vertically only. In all graphs, the scale for the x-axis is Ď&#x20AC;/2. The scale for the y-axis is listed above the graph. 1. (y-axis scale: 0.5)

2. (y-axis scale: 0.5)

3. (y-axis scale: 0.5)

4. (y-axis scale: 0.5)

5. (y-axis scale: 1/3)

6. (y-axis scale: 1/4)

Trigonometry/Pre-Calculus Graphing Sine and Cosine Functions – Homework

Name: ______________________ October 8, 2013

Part I. Graph each function over a two-period interval and give its amplitude. Be sure to label the axes! 1. y = 4 sin(x)

2. y = cos(x) − 1

1 3. y = sin(x) + 1 2

1 4. y = − cos(x) + 5 4

5. y = −3sin(x) + 1

6. y = 2 cos(x) + 3

Part II. Identify the functions graphed below. Each graph has been transformed vertically only. In all graphs, the scale for the x-axis is Ď&#x20AC;/2 and the scale for the y-axis is 1. 1.

Trigonometry/Pre-Calculus Graphing Horizontal Translations

Name: ______________________ October 9, 2013

Graph each function over a two-period interval. Be sure to label the axes! π 1. y = cos(x + ) 2. y = sin(x − π ) 2

1 3. y = cos( x) 2

5. y = 1+ sin(x +

4. y = −3 + sin(x)

π ) 2

6. y = −4 + 3cos(x −

π ) 2

8. y = sin(x + π )

7. y = 3 + sin(x)

9. y = cos(2(x −

11. y = sin(x +

π )) 4

π ) 2

10. y = 2 + cos(x)

12. y = cos(2(x +

π )) 4

Trigonometry/Pre-Calculus Graphing Horizontal Translations – Homework

Name: ______________________ October 9, 2013

Part I. Graph each function over a two-period interval. Be sure to label the axes! π π 1. y = sin(x − ) 2. y = sin(2(x + )) 2 8

3. y = cos(x + π )

5. y = 3 + sin(x +

4. y = cos(2(x −

π ) 2

π )) 8

6. y = 2 − cos(x +

π ) 2

Part II. Fill in the table below with the amplitude, period, vertical translation, and phase shift of each given function. Function 7. y = −4 + cos(x) 8. y = sin(x + π ) π 9. y = sin(2(x − )) 4 1 10. y = 3cos( x) 2 11. y = −1+ cos(x − π ) 1 12. y = −3 + 4 sin( (x − π )) 2

Amplitude

Period

Vertical Trans.

Phase Shift

Part III. Solve the problem. 13. The position of a weight attached to a spring is given by s(t) = −2 cos(7π t) , where s(t) represents the height, in inches, after t seconds. When does the weight first reach its maximum height?

14. A weight attached to a spring is pulled down 5 inches below the equilibrium position. Assuming that 7 the frequency is cycles per second, determine a model that gives the position of the weight at time t π seconds.

Trigonometry/Pre-Calculus Graphing Combinations of Transformations

Name: ______________________ October 10, 2013

Graph each function over a two-period interval. Be sure to label the axes! We will work through the first four examples together. The next four problems are all yours! π 1. y = 2sin(x − π ) + 1 2. y = − sin(2(x − )) 2

3. y = cos(x +

π )− 2 4

1 4. y = cos( x) + 1 2

π ) 4

5. y = −2sin(x) + 1

6. y = sin(x +

1 7. y = cos( x) + 3 2

8. y = −3cos(x − π )

Trigonometry/Pre-Calculus Graphing Horizontal Translations – Homework

Name: ______________________ October 10, 2013

Part I. Graph each function over a two-period interval. Be sure to label the axes! π π 1. y = sin(x − ) 2. y = sin(2(x + )) 2 8

3. y = cos(x + π )

5. y = 3 + sin(x +

4. y = cos(2(x −

π ) 2

π )) 8

6. y = 2 − cos(x +

π ) 2

Amplitude

Period

Vertical Trans.

Phase Shift

Trigonometry/Pre-Calculus Graphing Secant and Cosecant Functions

Name: ______________________ October 11, 2013

Graph each function over a two-period interval. Be sure to label the axes! We will work through the first six examples together. The next six problems are all yours! 1. y = csc(x) + 1

2. y = 2sec(x)

3. y = csc(2x) − 1

1 4. y = csc( x) 2

5. y = sec(x −

π ) 2

6. y = sec(2(x +

π )) − 1 4

π ) 2

7. y = − sec(x)

8. y = csc(x −

9. y = csc(−x) + 1

10. y = sec(π (x −

11. y = 3sec(x +

π )+1 4

π )) − 2 2

12. y = − csc(x + π ) + 1

Trigonometry/Pre-Calculus Graphing Secant and Cosecant Functions – Homework

Name: ______________________ October 11, 2013

Part I. Graph each function over a two-period interval. Be sure to label the axes! 1 π 1. y = − sin(x + π ) + 2 2. y = sin( (x + )) 2 2

3. y = −2 cos(x −

5. y = csc(−x)

π ) 4

4. y = cos(−x) + 3

6. y = csc(x + π ) + 3

7. y = 2sec(x −

π )+1 2

1 8. y = sec( x) − 1 2

Part II. Fill in the table below with the amplitude, period, vertical translation, and phase shift of each given function. Function 9. y = −3 − sec(x) π 10. y = 1+ sec(x − ) 2 11. y = − csc(2x) π 12. y = 1+ 2 csc(x − ) 2 13. y = −1+ sin(2(x + π )) 1 14. y = 3cos( x) 4

Amplitude

Period

Vertical Trans.

Phase Shift

Trigonometry/Pre-Calculus Graphing Tangent and Cotangent Functions

Name: ______________________ October 14, 2013

Part I. Graph each function over a one-period interval. Be sure to label the axes! 1. y = cot(x +

π ) 2

1 2. y = cot( x) 2

3. y = tan(x −

π ) 2

4. y = tan(2x)

5. y = cot(2x)

6. y = cot(x − π )

1 8. y = tan( x) 2

7. y = tan(x + π )

Part II. Fill in the table below with the amplitude, period, vertical translation, and phase shift of each given function. Function 9. y = 3cot(x) π 10. y = 2 tan(x + ) + 1 4 11. y = − tan(2x) π 12. y = − cot(x − ) + 3 4

Amplitude

Period

Vertical Trans.

Phase Shift

Part III. Solve the problem.

π 13. The position of a weight attached to a spring is given by y = − cos( t) + 2 , where t represents the 2 time, in seconds, and s(t) represents the position, in inches, at time t. When does the weight first reach its maximum height?

14. A weight attached to a spring is pulled down 3 inches below the equilibrium position. Assume the 2 frequency is cycles per second. Determine a function s(t) that gives the position of the weight at time t π seconds.

Identification of Subgroups The following subgroups were identified for this unit: Female Students There is a well-documented mathematics achievement gap nationally among both female students relative to their male peers and minority students relative to their white peers. While there is only one minority among the 16 students in this class, making this subgroup impractical to study, the gender split in the class (nine females, seven males) makes this a perfect case study. In addition, this group is heterogeneous with respect to students’ mathematical abilities—of the nine members, two received grades of “A” in their most recent math class, four received a “B,” and three received a “C,” giving it an ideal mix. Members of group: Students 1, 2, 5, 6, 8, 11, 13, 14, and 16. Students Scoring 80% or Below on Most Recent “Speed Test” Throughout this marking period, students have been given three “speed tests” designed to measure their knowledge of quadrantal trigonometric values. Students have been made aware repeatedly that the only way they will be successful in this class is by committing these values to memory, and the speed tests serve as an ongoing review and remediation. On the most recent speed test, seven students scored 80 percent or lower, suggesting a continuing lack of command of the trigonometric values. Although this unit involves graphing trigonometric functions—an activity for which, in theory, the respective values must be known—it is possible students can succeed in this unit simply by memorizing the starting points of the graphs as well as their respective patterns. This subgroup will be examined to determine whether the latter strategy will lead to successful outcomes, or whether these students will struggle in the unit due to the gap in their knowledge base of the fundamentals underpinning the material. Members of group: Students 1, 3, 4, 5, 10, 11, and 13.

Section 4: Lesson Plans

Pretest Item Analysis (Whole Class) Grade: 12 Objectives:

Subject: Trigonometry/Pre-Calculus Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5

Unit: Trigonometric Functions

Intern: Ben Cohen

Graph the sine and cosine functions, including vertical translations, and identify amplitude and period. Graph horizontal translations of the sine and cosine functions and identify period and phase shift. Graph combinations of translations of the sine and cosine functions. Graph the cosecant and secant functions, including horizontal and vertical translations and combinations. Graph the tangent and cotangent functions; solve problems involving simple harmonic motion. PRETEST ASSESSMENT ITEMS

STUDENT

ALIGNMENT WITH LESSON OBJECTIVES

X X

1 2

3 4

6 7 8 9

X X X

11 12 13 14

X X X

X 8

X 4

15 16 TOTAL CORRECT RESPONSES

Key$%$X$indicates$correct$response$

TOTAL CORRECT RESPONSES

1 2 0 0 2 0 2 1 2 2 0 2 2 1 0 2 19

Instructional Implications Based on Analysis of Pretest Data (Whole Class) •

The area at which students were most successful was identification of a vertical translation; 50 percent of students (eight out of 16) correctly identified this on the pretest. While this is not nearly a large enough percentage to indicate class-wide mastery of the skill, it does show that they have some familiarity with vertical shifts, which were part of their Algebra II curriculum. Although it will still be necessary to treat this as entirely new material, I will be able to move slightly more quickly through this than if students had no prior knowledge of it. The CCPS curriculum guide suggests devoting two days to vertical shifts, but I expect to be able to compress this into one lesson, freeing up instructional time to focus on applications of these functions.

•

Twenty-five percent of students (4/16) were able to successfully identify a phase shift on the pretest. Once again, this reflects content knowledge that they should have acquired during Algebra II; however, it is likely that some students were confused by the use of the term “phase shift” (which is used with periodic functions) rather than “horizontal shift,” which applies to aperiodic functions. Again, as students have some basic knowledge of this material, I will likely be able to cover it in one lesson rather than the two suggested by the CCPS curriculum guide.

•

Fourteen out of 16 students (88 percent) were unable to determine the period of a given function; the same percentage was not able to find the amplitude of a function. This is not surprising, since students have had no prior exposure to periodic functions, and these two concepts do not apply to aperiodic functions. I will use students’ prior knowledge of symmetry to help them understand the concept of a period; in addition, as it is likely that the students who were able to correctly identify these items learned them from a previous physics class, they will be introduced not in a vacuum, but as part of a cross-curricular application so that all students may benefit from this exposure.

•

Only one student was able to correctly graph a sine and cosine function; no students were able to graph a secant or cotangent function. Although students are expected to demonstrate a thorough knowledge of trigonometric values at this point in the semester, they likely do not know how to translate these values into a graph. Graphing instruction will make use of students’ existing knowledge of trigonometric values.

•

Only one student was able to solve a problem involving simple harmonic motion, and none were able to determine a model for such a problem. Three students did indicate that they know what needs to be done (e.g., “solve for t”); it appears they simply do not know how to do so. When teaching this portion of the unit, I will show students how to work backwards to find the solution, making use of and extending the existing content knowledge of these students while equipping the other 13 students with the tools they will need to succeed on these types of problems.

Pretest Item Analysis Disaggregated Subgroup â&#x20AC;&#x201C; Female Students Grade: 12

Subject: Trigonometry/Pre-Calculus

Objectives:

Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5

Unit: Trigonometric Functions

Intern: Ben Cohen

STUDENT

ALIGNMENT WITH LESSON OBJECTIVES

X X

1 2

4 5

6 7 8 9 TOTAL CORRECT RESPONSES

X X X 5

! Key$%$X$indicates$correct$response$

! !

X X 3

TOTAL CORRECT RESPONSES

1 2 2 0 1 0 2 1 2 11

Instructional Implications Based on Analysis of Pretest Data (Disaggregated Subgroup: Female Students) •

Much like the class as a whole, female students were most successful at determining the vertical shift of a function; 56 percent of students (five out of nine) successfully identified this. As with the class as a whole, it is good to see these students have retained this content knowledge from Algebra II. In addition, considering the heterogeneous nature of this group with respect to students’ achievement, it was gratifying to see that students’ mastery of this topic exceeded that of the class as a whole (50 percent). Based on this data, no instructional modifications are necessary for this portion of the unit.

•

Thirty-three percent of female students (3/9) were able to successfully identify a phase shift on the pretest. Once again, this was higher than the percentage of the class as a whole (25 percent). This also reflects content knowledge that students should have acquired during Algebra II, though some were likely confused by the use of the term “phase shift” rather than “horizontal shift,” which was presumably used during their study of aperiodic functions. Again, as students have some basic knowledge of this material, it appears no modifications are necessary for this particular subgroup.

•

Seven out of nine students (78 percent) in this subgroup were unable to determine the amplitude of a given function, while eight out of nine (89 percent) could not determine the period; this again closely parallels the whole-class data (88 percent for each). Since these concepts will be introduced as part of a cross-curricular application, however, some management concerns must be taken into consideration. For example, sports analogies should not be limited to male-centric sports such as football; this subgroup includes members of the lacrosse and track and field teams, so examples should be drawn from these sports as well. Similarly, music examples should not include only guitar solos from all-male bands; a clip from female-centric band Heart should be included as well.

•

No female students were able to successfully graph a trigonometric function; as only one student in the whole class was able to graph a sine and cosine function (and none were able to graph a secant or cotangent function), this does not represent a significant departure from the whole-class patterns and it does not appear that any special instructional procedures are necessary for this topic.

•

No students in this subgroup were able to solve a problem involving simple harmonic motion or determine a model for such a problem. Only one of the three students who indicated that they know what needs to be done (e.g., “solve for t”) is a member of this subgroup (Student 2). Although there will be no difference in instructional procedures for female students during this portion of the unit, care must again be taken to ensure gender neutrality in examples to the fullest extent possible—such as using the height of a volleyball during a point rather than a thrown or kicked football.

Pretest Item Analysis Disaggregated Subgroup – 80% or Less on Last “Speed Test” Grade: 12

Subject: Trigonometry/Pre-Calculus

Objectives:

Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5

Unit: Trigonometric Functions

Intern: Ben Cohen

STUDENT

ALIGNMENT WITH LESSON OBJECTIVES

1 2 3

6 7 TOTAL CORRECT RESPONSES

X 1

! Key$%$X$indicates$correct$response$

! !

X 2

TOTAL CORRECT RESPONSES

1 0 0 2 2 0 2 7

Instructional Implications Based on Analysis of Pretest Data (Disaggregated Subgroup: 80% or Below on Last “Speed Test”) •

As expected, students in this subgroup struggled mightily relative to their peers on the pretest; nine of the 10 questions were answered correctly by fewer than two students, although three students correctly answered more than one question. The lone exception was question 2, which asked students to identify the phase shift of a function; two out of seven students (29 percent) correctly answered this question. Interestingly, only four students in the entire class correctly answered this question. As with the class as a whole, this likely reflects content knowledge from Algebra II; it appears this group will not need any special remediation for this topic.

•

However, only one member of the subgroup (Student 13) was able to successfully identify a vertical translation; the other student who correctly identified a phase shift (Student 5) appeared to mix up period and vertical translation. This again reflects the differences in students’ content knowledge from Algebra II; although I will have to introduce this as completely new material, it is not necessary to know trigonometric values to perform a vertical translation, so I expect these students to be able to master this topic in one day rather than two, as suggested in the CCPS curriculum guide.

•

One student out of the seven in this subgroup (14 percent) was able to identify the period and amplitude of a given function. Unlike with vertical translations, students will need to know the respective values of the function to determine these values; as such, the students’ struggles here were expected. These topics may have to be covered in tandem with some remediation of trigonometric values; I will attempt to make this as studentcentric as possible, such as by asking students to provide the values for quadrantal angles, rather than making it simply a rote review.

•

Similarly, no students in this subgroup were able to graph a cotangent or secant function; one student did, however, successfully graph a sine and cosine function. Interestingly, this student (Student 10) received the second-lowest grade in the class on the most recent “speed test” (17/48, or 35 percent) and was also the last student to finish his pretest, suggesting his struggles may be rooted more in an inability to rapidly recall the information rather than a lack of command of the material. As this may apply to other students in this group, I will need to be cognizant that these students may desire or require additional assistance after I demonstrate how to graph a function; example problems will come from the classwork, so this will be addressed during the individualwork portion of the lesson.

•

No students in this subgroup were able to solve a problem involving simple harmonic motion or determine a model for such a problem. Although this was not unexpected for any groups of students, this subgroup will likely encounter more difficulty with these problems given their lack of command of trigonometric values; solving a simple harmonic motion problem requires knowledge of the angles at which the sine and cosine functions reach their maximum and minimum values. I must thus keep in mind that I may need to build in a remediation component when explaining how to solve these problems.

Instructional Lesson Plan I. Purpose of the Lesson – What will the students learn? Why is this learning meaningful, important and appropriate? What will the students say or do that will serve as evidence of learning? Subject Trigonometry/Pre-Calculus Date October 8, 2013 Mentor Mark Heffner Standard(s): ____ CCSS

Unit Graphing Trigonometric Functions Grade Class Size 12 16 School Winters Mill High School

Topic Sine and Cosine Functions Time 1:00-2:20 p.m. Intern Ben Cohen

Other: Maryland Content Standards

2.1.2 The student will identify the domain, range, the rule or other essential characteristic of a function. 2.2.3 The student will perform translations, reflections, and dilations on functions. Big Idea or Essential Question Up to this point, students have worked only with aperiodic functions—those that do not repeat their values in any sort of pattern or with regularity. This lesson will serve as their introduction to periodic functions, those that do repeat in a set pattern, or period. Students will begin by working with the graphs of the sine and cosine functions, topics with which they are already familiar from previous units in this course. They will also learn how to apply vertical transformations to these functions—a topic with which they should be familiar from Algebra II, but one that they have never previously applied to periodic functions.

Alignment with Summative Assessment Question 1 on the summative assessment asks students to determine the vertical translation of a cosine function; question 4 asks students to determine the amplitude of a sine function. Lesson Objective Students will be able to graph the sine and cosine functions, including vertical translations, and will be able to identify the amplitude and period of these functions. They will also be able to identify the vertical translations in either a given function or its graph. Formative Assessment Following the lecture portion of the lesson, students will complete a 12-question problem set in class. Students will also complete an additional 12 questions for homework. All questions will ask students to graph sine and cosine functions, including vertical transformations (shifts, stretches, and/or compressions), over a two-period interval, or identify the graphs of vertically transformed functions. They will also be asked to identify the amplitude of sine and cosine functions.

II. Context for Learning – What knowledge of students will influence my instructional decisions in this lesson? How will my instruction remove barriers to learning and/or build on students’ strengths? Knowledge of Learners

►►►►►

(What prior knowledge of learners are you using to plan this lesson? DATA

Instructional Decisions Based on Knowledge of Learners RESPONSE TO DATA (required)

Formal data - Pretest, formative assessment, checklists, etc.

• Exactly half of students (eight out of 16) scored a 90% or higher (43/48) on the most recent “speed test,” designed to measure their knowledge of trigonometric values. Four students failed to score above a 75; all four scored below 50% (Student 3 scored a 10/48, or 21 percent; Student 4 scored a 22/48, or 46 percent; Student 5 scored a 20/48, or 42 percent; and Student 10 scored a 17/48, or 35 percent). One student (Student 14) scored 100%; two others (Students 15 and 8) scored 47/48.

• Although the expectation at this point is that students will have quadrantal trigonometric values committed to memory, it appears some students may still benefit from warm-up and other activities involving nothing more than recalling these values. It may also be the case that some students may simply need more time to think of the values, especially since one of the four students who scored below 50% on the assessment (Student 5) has otherwise performed strongly in the class, so I must allot adequate time for these students to complete the classwork since the use of calculators will not be allowed on assessments.

Informal data - Observations of students, reflections from previous lesson(s), anecdotal records, etc.

• Students are expected to be familiar with vertical transformations of functions from their Algebra II curriculum (all students have taken Algebra II prior to this course).

Individual or Small Group Needs Specific to this Lesson

►►►►►

(Ex. IEP/504 accommodations, ELL, social concerns, gifted/talented, multicultural/equity measures, etc.)

• Student 4 has a 504 plan due to hearing loss.

• Although students have previously seen function transformations, it is not a guarantee that they will recall all of the previous material for this topic. As such, I must be flexible when it comes to this— students may require more time and/or practice problems than I had originally allotted, or they may instantly recall this and not require as much practice as I had originally anticipated. This is particularly applicable given the range of student achievement in the class—of the 16 students, six received an A in Algebra II, six received a B, and four received a C. As such, the number of problems on the formative assessments that the class will go through together is approximate and may change as the class’s needs dictate.

Differentiated Practices Specific to this Lesson (Instruction and/or Assessment)

In addition, I will need to repeat questions posed by students to the whole class so he can understand (e.g., “Allison just asked…”). • Care must be taken to include a cross-curricular component in the lesson and to engage all learners, being mindful of the gender split and high percentage of FARMS students.

• Cross-curricular connections will be as universally applicable as possible—for example, the “realworld” application of a roller coaster modeling a sine wave will not be used, since some students may have not had the opportunity to ride a roller coaster and thus would not be able to relate to this. Instead, they will involve easily recognizable items such as a person’s heartbeat.

Technology Integration All students will use the TI Nspire calculators in the classroom to model trigonometric functions and their vertical transformations. This will only be done to aid students’ understanding of the concepts; students will be required to use pencil-and-paper methods on the formative assessments. The instructor will also prepare a PowerPoint to help demonstrate the new concepts and terms in this lesson; the PowerPoint will also include practice problems that can be worked out in front of the entire class using the ActivInspire tablet, which allows the user to annotate the screen. Cross-curricular Connections Students will be shown how to model problems using the sine and cosine functions and their vertical transformations. Examples will be drawn from various fields, including physics (simple harmonic motion, the example given in the CCPS curriculum guide), health (using a sine wave to model blood pressure), meteorology (using sine and cosine functions to model the high and low temperatures for a given region), music (guitar notes), and sports (using the cosine function to model the height of a volleyball during a match). Students will be exposed to these applications, but will not be expected to solve such problems until the following lesson, when they will also have knowledge of horizontal transformations.

• The TI Nspire calculators used during this lesson are the property of the school and remain in the classroom at all times; students are not permitted to bring them home for work or practice. In addition, given the high percentage of FARMS students at the school, all students cannot be expected to own a graphing calculator of their own. As such, the instructor must be prepared to devote extra time to showing students tips and tricks for graphing these functions on the calculator, since unlike a TI-83, they are still learning how to use these calculators.

Instructional Sequence Planned Beginning • • •

Approximate Time 15 minutes

Warm-up Motivation Bridge

Procedure • Students will begin with a brief pre-test designed to measure their understanding of the prerequisite knowledge and objectives for this unit (function transformations). • Following this, student knowledge will again be verified by a warm-up activity (tossing around a small football) that asks students to provide values for quadrantal angles (e.g., “cosine of zero”). As the answers are provided, plot them on the coordinate plane at the front of the room—the beginning of the graph of the sine function.

Development of the New Learning (Procedure will vary with the instructional model used. Ex. 5E lesson plan, scientific inquiry, teacher/student modeling, cooperative learning, before, during and after strategic reading, etc. Focus on active student engagement.)

35 minutes

• Trigonometric functions represent students’ introduction to periodic functions—that is, those that repeat over a specified interval. After graphing values of sine from 0 to 2π, ask students about values beyond the one period (e.g., -π or 3π). Explain to them that sine is an example of a periodic function, and that the pattern they observe repeats infinitely. • Using a different color, graph values of cosine from 0 to 2π (again surveying the students for values). As with the sine function, ask students about values beyond the one period and note that the cosine function is another example of a periodic function. • Use the TI Nspire calculators to show students the graph of both the sine and cosine functions (graphs will have been previously loaded onto all students’ calculators). Using the emulator on the instructor’s computer, project the same image to the class. • Using a PowerPoint, demonstrate to students some examples of periodic functions, such as sound waves and a heart monitor. Pose the question to students: How can the graph of the sine function match up with one of these examples? • Remind students of the definition of a function: a map from one set to another. Using the PowerPoint, demonstrate with a table of values the mappings of the sine and cosine functions. Now raise the question as to what happens if these values are transformed—for example, adding one to each of them, or multiplying them by two. Similarly, how would this affect the graph of each function? Demonstrate this for the students.

Enrichment or Remediation

25 minutes

(As appropriate to lesson)

• Introduce students to the concepts of vertical transformations. The first graph will be sin(x) + 2 (a vertical translation); the second will be 3cos(x) (a vertical stretch). Students will not be given the equations for these functions; rather, the instructor will help them discover the equations. Introduce the definition of amplitude: half the difference between the maximum and minimum values of a function. Illustrate this on the graph as well. • Distribute a worksheet with 12 questions on it. Six will ask students to draw a transformed sine or cosine graph; the remaining six will ask students to determine the equation of a graphed function, and the remaining four will ask students to determine the amplitude and/or shift of a given function. The class will do four examples together (two of each type of problem), after which students will complete the rest of the worksheet individually. • Circulate as students are working to answer any questions and informally assess students’ understanding of the material.

Assessment/ Evaluation

Ongoing

• Students will be informally assessed throughout the lesson. During the lecture component, the instructor will look around the room to see if any students are visibly having difficulty understanding. As students are working individually on their worksheets, the instructor will circulate to answer questions and ensure all students comprehend the material and are remaining on topic. • Take time throughout the lesson to ask if any students have questions about the material. • Additional formative assessment: For homework, students will complete a 12-question problem set similar to the one given in class. Six questions will ask them to determine the graph of a transformed/translated function, while the remaining six will ask them to identify the function shown in a graph.

Planned Ending (Closure) • •

Summary Homework

5 minutes

• Briefly summarize and restate the information that was covered during the lesson. • Take time to see if any students have remaining questions about the material. • Pass out homework (handout with 12 problems).

• Two out of 16 students (Students 15 and 12) correctly completed all 12 problems on the formative assessments; one more (Student 14) correctly completed 11 out of 12. Six out of 16 successfully completed seven or few problems.

What trends or patterns do you notice that indicate strengths and/or areas of need for the class as a whole, subgroups or individuals?

• All students (save for Students 15 and 12) struggled with the most complicated problem on the assessments, which involved a combination of both a vertical shift and amplitude change. Many also struggled with problems involving a vertical compression (i.e., fractional amplitude); 13 out of 16 students were not able to graph these functions accurately.

What will be your next steps based on this analysis?

• I will build in review time for vertical transformations at the start of the next lesson before I introduce new material (horizontal transformations). I will make sure at least 75 percent of the class (12/16) is comfortable moving forward before I end the review and begin introducing the new material. Reflection What instructional and/or assessment activities were effective? Why? What instructional and/or assessment activities were less effective? Why? What changes would you make to your instructional and/or assessment procedures that would improve student learning? How did you “think on your feet” to make instructional decisions while you were teaching the lesson? What worked or did not work about these decisions? How did the oral or written feedback you gave students address their needs in relation to meeting the lesson objective? What multicultural considerations emerged during this lesson that you had not anticipated? How would you address those in the future? What insights have you gained from teaching this lesson?

• Students thoroughly enjoyed the cross-curricular applications included in the lesson; this proved to be an even more effective tool than I had hoped. I will continue to include these in subsequent lessons to foster student engagement and understanding. In addition, I will move these from midway through the lesson to the end, after the new material has already been introduced, so that students can experience the full benefit. • One of the activities I had planned for this lesson—determining a function based on its transformed graph—proved to be too advanced for the students given their knowledge. In retrospect, it would have been wiser to introduce this activity on a later day rather than on the first day to ensure students had developed a sufficient understanding of graphing functions and their transformations; the first day should have been devoted entirely to graphing a function based on the transformations given, and not the other way around. Both the in-class formative assessment and homework assignment originally featured problems similar to this, but students were told to disregard these problems given the difficulties they demonstrated with them (although a few students were able to successfully complete this task). In addition, the assessments for the following lesson were redesigned to include only graphing questions rather than a graph-identification component. • FLEX time in the morning was cut short by an unplanned fire drill. Although this did not directly affect my lesson, it did get me thinking about contingency plans in case lessons are interrupted or cut short by circumstances beyond my control (e.g., emergency drill, utility failure, early dismissal due to snow, nuclear attack). In much the same way it is crucial to prepare for a lesson ending too early, I must be sure to take this into account when planning a unit.

V. InTASC or COE Standard and Rationale - How does this lesson represent my progress toward gaining mastery of the skills of an effective teacher? This lesson is a good example of InTASC or COE ____ because . . . • InTASC 1 (Learner Development): Although this is a high-achieving class, there are still some students who occasionally struggle with basic concepts such as the values of quadrantal angles; at the other end of the spectrum are students who, though not formally identified as gifted and talented, achieve at a level higher even than that of their peers. This lesson has been designed with both of these groups of students in mind, with warm-up activities designed to both introduce the new material and simultaneously activate students’ knowledge of the prerequisite skills they will need to be successful and cross-curricular connections that provide both another way of looking at the material for the lower-achieving students and a challenge for the higher-achieving ones. In addition, the examples used take into consideration the high percentage of FARMS students at the school; all are everyday situations to which all students can relate, such as the weather or objects in the night sky. • InTASC 5 (Application of Content): This lesson includes an introduction to the myriad applications of trigonometric functions, with applications in fields as disparate as health, music, astronomy, sports, and meteorology. Many of these, such as high and low temperatures and blood pressure, represent situations to which students can legitimately relate—for example, every student in the class has experienced the high and low temperatures of a given day (including some that vary greatly). Opportunities for students to brainstorm their own applications of a trigonometric function are prevalent not just in this lesson, but throughout the unit; these cross-curricular connections are intended to help students engage in critical thinking and gain a greater appreciation for the enormous impact these seemingly irrelevant topics have on their everyday lives. • InTASC 8 (Instructional Strategies): Although this lesson is more heavily lecture-based than the others, since it is the introduction to the unit, I have employed a number of different instructional strategies to ensure students remain engaged throughout the lesson. Opportunities for student participation are particularly prevalent throughout the PowerPoint; while this may seem like little more than a tool for “spicing up” the lecture, in fact there are several junctures at which point students are encouraged to think deeply about the material being presented. For example, three guitar solos will be used to illustrate the applications of the sine function; students will also be asked to identify the transformations illustrated by a series of graphs, in addition to graphing transformations of functions. This lesson includes components designed to appeal to all types of learners, with the PowerPoint serving not as a crutch for the lecture, but as a means for activating these components.

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Sine Waves Trigonometric Functions October 8, 2013

!  As you are all aware, the sine function repeats at

values beyond 2π—for example, sin(π/2) = sin(5π/ 2) = sin(9π/2)…

!  The sine function is thus an example of a periodic function (also called circular function), since it repeats over a set period—in this case, 2π.

!  More broadly: A periodic function is a function f(x)

for which f(x) = f(x + np) for all x in the domain of f, every integer n, and some positive real number p, called the period.

Sine Waves

Cosine Waves

Notice anything?

So when will I use this? !  Trigonometric functions have applications in

numerous fields, including: !  Physics (simple harmonic motion, e.g., weight on a

!  !  !  !  !

spring) Health (blood pressure graph) Meteorology (high and low temperatures/tides) Astronomy (position of Pluto relative to planets) Sports (height of a volleyball during a match) Music (sound waves)

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Example 1: Tides

Example 2: Sound waves from a guitar

Example 3: Blood pressure

Try it out! !  Thanks to his gene pool, Mr. Cohen’s blood

pressure is 130/72. Let’s model this using a trigonometric function!

!  WAIT! The maximum and minimum values of sine and cosine are 1 and -1. How can we change the graph so that its new maximum and minimum values are 130 and 72??

Function transformations !  Remember the definition of a function: A relation

that gives each input a unique output. A graph is NOT a function, but rather the visual representation of a function.

!  The graphs you were shown earlier were not the sine and cosine functions, but their visual representations.

Function transformations Function sin(x) 2sin(x)

0 0 0

π/2 1 2

π 0 0

3π/2 -1 -2

2π 0 0

!  In a function of the form f(x) = a*sin(x) or f(x) =

a*cos(x), |a| is known as the amplitude. A value of |a| greater than 1 results in a vertical stretch, while a value of |a| less than 1 results in a vertical compression.

!  Negative values of a will reflect the function across the x-axis, in addition to any stretch/compression.

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Function transformations

Function translations

Vertical shifts

Function cos(x) cos(x) + 1

0 1 2

π/2 0 1

π -1 0

3π/2 0 1

2π 1 2

!  In a function of the form f(x) + c, c is known as the vertical shift or vertical translation. The function will move up (c > 0) or down (c < 0) the specified number of units.

!  This is applicable to all functions, not just sine and cosine!

Vertical shifts

Revisiting the earlier example… !  Amplitude changes and vertical shifts can be

combined. Let’s revisit the earlier example of modeling Mr. Cohen’s blood pressure.

!  Since it starts at the maximum value (systolic pressure), we will need to use a cosine wave.

!  This cosine wave will need to be both stretched and shifted to fit the criteria.

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Revisiting the earlier example…

!  How do we know what the amplitude of the stretched function must be?

!  Amplitude is equal to half the value of the maximum and minimum values of a periodic function—in this case, ½(130 – 72) = 29.

!  So we know the function will take the form 29cos(x) + c. What will be the value of c?

!  The maximum and minimum values are currently 29 and –29. 130 – 29 = 101; 72 – (–29) = 101.

!  So our function is 29cos(x) + 101.

Identify the function!

Practice time!

Instructional Lesson Plan I. Purpose of the Lesson â&#x20AC;&#x201C; What will the students learn? Why is this learning meaningful, important and appropriate? What will the students say or do that will serve as evidence of learning? Subject Trigonometry/Pre-Calculus Date October 9, 2013 Mentor Mark Heffner Standard(s): ____ CCSS

Unit Trigonometric Functions Grade Class Size 12 16 School Winters Mill High School

Topic Horizontal Translations Time 1:00-2:20 p.m. Intern Ben Cohen

Other: Maryland Content Standards

2.1.2 The student will identify the domain, range, the rule or other essential characteristic of a function. 2.2.3 The student will perform translations, reflections, and dilations on functions. Big Idea or Essential Question Yesterday, students were introduced to periodic functions and their vertical transformations. They also learned that these functions can be used to model problems in fields as disparate as health and meteorology. Today, we will continue our study of transformations of the sine and cosine functions with a look at horizontal transformations. In addition to giving students new insight into the relationship between the two functions, this will equip students with the tools necessary to model and solve problems such as those introduced in the previous lesson.

Alignment with Summative Assessment Question 8 on the post-test asks students to graph a cosine function given a horizontal translation (8); questions 2 and 3 ask them to find the phase shift (2) and period (3) of a function. Questions 9 and 10 ask students to solve word problems concerning simple harmonic motion utilizing sine and cosine functions. Lesson Objective Students will be able to graph horizontal translations of the sine and cosine functions; they will also be able to identify the period and phase shift of these functions. Students will also be able to use these functions to solve problems involving simple harmonic motion. Formative Assessment Following the lecture portion of the lesson, students will complete a 12-question problem set in class. Students will also complete an additional 14 questions for homework. All 12 questions on the classwork and six on the homework will ask students to graph sine and cosine functions, including both vertical and horizontal transformations; six more on the homework will ask them to identify the amplitude, period, and phase shift of a given function. In addition, on the homework, there will be two word problems asking students to model given situations using trigonometric functions and their transformations.

►►►►►

(What prior knowledge of learners are you using to plan this lesson? DATA

Instructional Decisions Based on Knowledge of Learners RESPONSE TO DATA (required)

Formal data - Pretest, formative assessment, checklists, etc. • Only 25 percent of students (4/16)—Students 5, 9, 16, and 13—successfully identified a phase shift on the unit pre-test. Three students (Students 5, 7, and 2) successfully determined the period of a function.

• While the phase shift data are not entirely unexpected, this does reflect content knowledge that students should have retained from Algebra II. If student achievement had been higher, I would plan to simply review this; however, as it is clear that the majority of students lack this knowledge, I will teach it to the class as though they were learning it for the first time. I will adjust my instruction accordingly if this successfully activates their knowledge of the topic. Students are not expected to know how to determine the period of a function since this is not something they have studied previously; the data confirm that I should not change my strategy of introducing it as entirely new material.

Informal data - Observations of students, reflections from previous lesson(s), anecdotal records, etc.

• This is the first time that students will have experienced word problems utilizing the trigonometric functions; all of their previous experience with word problems in this class has involved determining the side length or angle measure in a triangle. Modeling problems is entirely new to them.

• Currently, I plan to work through two examples with the class as a whole. However, I need to have several other examples on hand to go through as a class to accommodate the students if they would like more practice.

• Students struggled with the second planned activity, identifying a function based on its graph. The intent of this activity was to get them to work backwards in terms of function transformations, but they had not yet acquired sufficient knowledge to complete the activity when I introduced it.

• I had initially planned to include a similar activity during this lesson, but based on students’ difficulties with a simpler version, I have scrapped this in favor of an increased focus on graphing functions as well as identifying their key features. This activity may be reintroduced later in the unit if I feel students have acquired sufficient knowledge to allow them to succeed.

• Students thoroughly enjoyed the cross-curricular activities I included in the lesson. These were designed to be as relevant and interactive as possible.

• While this lesson includes a number of cross-curricular connections as well, some of them (such as simple harmonic motion) likely model concepts that students will find either uninteresting or irrelevant. I will introduce additional connections—such as observing the sound waves generated by music of differing frequencies—that provide more opportunities for student engagement.

Individual or Small Group Needs Specific to this Lesson

►►►►►

(Ex. IEP/504 accommodations, ELL, social concerns, gifted/talented, multicultural/equity measures, etc.)

• Student 4 has a 504 plan due to hearing loss.

Differentiated Practices Specific to this Lesson (Instruction and/or Assessment)

• He is seated in the front of the classroom during

all class meetings (students’ seats were assigned at the start of the academic year). I will need to speak slightly louder than usual during this class, both during lecture and as I am circulating the classroom as students are working individually. In addition, I will need to repeat questions posed by students to the whole class so he can understand (e.g., “Allison just asked…”). • Care must be taken to include a cross-curricular component in the lesson and to engage all learners, being mindful of the gender split and high percentage of FARMS students.

• Cross-curricular connections will be as universally applicable as possible and will involve items that all students will have had the chance to experience. In this instance, the relationship between frequency will be modeled using images of the sound waves determined by the WMHS course bell and fire alarm, since all students will have had equal access to, and be familiar with, these items.

III. Instructional Procedures – What instructional strategies will I use to ensure that every child is a successful learner? Instructional Materials/Resources A handout will be used in class that contains problems for the students to work out, working individually; a second handout will be prepared for homework. Both handouts will include spaces for students to graph the functions, rather than using a separate handout with a graph on it. In addition, students will be given notecards reading “YES” and “NO” that they will use to answer questions from the instructor regarding their mastery of the material and whether he can proceed or if more practice is needed. This lesson will also utilize a PowerPoint presentation and the TI Nspire calculators provided in the classroom (detailed below).

Technology Integration All students will use the TI Nspire calculators in the classroom to model trigonometric functions and their horizontal transformations. This will only be done to aid students’ understanding of the concepts; students will be required to use pencil-and-paper methods on the formative assessments. The instructor will also prepare a PowerPoint to help demonstrate the new concepts and terms in this lesson; the PowerPoint will also include practice problems that can be worked out in front of the entire class using the ActivInspire tablet, which allows the user to annotate the screen. Cross-curricular Connections Students will be shown how to model problems using the sine and cosine functions and their horizontal transformations; unlike in the previous class, when students were only shown that it is possible to model these problems, students will now have enough knowledge to solve these problems. Examples will be drawn from the fields of physics, health, meteorology, and sports, as was done in the previous class; in addition, there will be multiple examples within fields (for example, physics examples will include both simple harmonic motion and energy).

students will have access to the calculators for the duration of the class, in case it becomes necessary to adjust the lesson plan “on the fly” and give them extra time to practice graphing functions with the calculators. In addition, classwork will be passed out at the start of class, as students are working on the warm-up, to ensure instructional time is not wasted while distributing this material. • Opportunities for student engagement are included throughout the lesson. Students will be asked to volunteer answers to the warm-up questions, the differences between shifted graphs, and values of sine and cosine for quadrantal angles, as well as helping work out word problems that will be posed during the enrichment portion of the lesson. In addition, during the individual-work portions of the lesson, the instructor will circulate to actively engage students and ensure they are staying on task. • The TI Nspire calculators used during this lesson are the property of the school and remain in the classroom at all times; students are not permitted to bring them home for work or practice. In addition, given the high percentage of FARMS students at the school, all students cannot be expected to own a graphing calculator of their own. As such, the instructor must be prepared to devote extra time to showing students tips and tricks for graphing these functions on the calculator, since unlike a TI-83, they are still learning how to use these calculators.

Instructional Sequence Planned Beginning • • •

Approximate Time 10 minutes

Warm-up Motivation Bridge

Procedure • For their warm-up activity, students will graph three functions that have been vertically transformed. When they have finished, ask a student to come to the board and graph one of the functions; ensure all students understand the answer. This will be repeated for all three functions. • Yesterday, all of the transformations were applied outside of the parentheses—or, using the definition of a function and our value tables, to the outputs. What happens if we now apply a transformation inside the parentheses (that is, to the inputs)?

35 minutes

• Begin by reminding students of the basic idea behind functions: a function, at its core, maps a set of elements to another set of elements. For example, sin(x) maps 0 to 0, π/2 to 1, π to 0, 3π/2 to -1, and 2π to 0 (and so on). Illustrate this using a table of values, as was done in the previous lesson. • Suppose now we wanted to model the function sin(x + π/2). Illustrate using a table of values how this would change the mapping: f(0) is now 1 rather than 0, f(π/2) is now 0 rather than 1, f(π) is now -1 rather than 0, and so on. Now, compare the graphs of the two functions (draw on the coordinate plane using two different markers). Note to students that the original graph of sin(x) has been shifted to the left π/2 places. (Emphasize that it may appear counterintuitive—positive values shift the graph to the left and negative ones to the right.) Inform students that this is known as a phase shift. • Do the same with the graph of the cosine function, except with a negative phase shift (e.g., cos(x-π)). Once again, demonstrate this first with a table of values and then with a hand-drawn graph. • Use the TI Nspire calculators to show students the phase-shifted

graphs of both the sine and cosine functions. Using the emulator on the instructor’s computer, project the same image to the class. Enrichment or Remediation

30 minutes

(As appropriate to lesson)

• Introduce students to the concept of period changes, again using a table of values. Begin by graphing cos(2x)—how does this graph compare to the graph of cos(x)? What about sin(-x) and cos(-x)? • Again, use the Nspire calculators to show students graphs of sine and cosine functions with period changes. Project the images to the class as well. • Revisit some of the models covered in the previous lesson. Students will now have the knowledge necessary to run through these models. Guide students through two examples of word problems that can be solved using their newly acquired knowledge. • Begin a worksheet with 12 functions on it (phase shifts and/or period changes involving sine and cosine functions). The class will do the first four examples together, after which students will complete the rest of the worksheet individually. All 12 questions will ask students to graph the given function. • Circulate as students are working to answer any questions and informally assess students’ understanding of the material.

Assessment/ Evaluation

Ongoing

• Students will be informally assessed throughout the lesson. During the lecture component, the instructor will look around the room to see if any students are visibly having difficulty understanding. As students are working individually on their worksheets, the instructor will circulate to answer questions and ensure all students comprehend the material and are remaining on topic. • Take time throughout the lesson to ask if any students have questions about the material. • Additional formative assessment: For homework, students will complete a 14-question problem set similar to the one given in class. Six questions will ask them to determine the graph of a transformed/translated function, while six more will ask them to identify the period and phase shift of a given function. The final two problems will be word problems asking students to translate a trigonometric function to model a given scenario.

Planned Ending (Closure) • •

Summary Homework

5 minutes

IV. Analysis and Reflection – To what extent are my students learning? How have I improved my professional skills? Analysis What does the data from the formative assessment indicate about the extent to which students acquired the intended learning? (Cite specific numbers or percentages of students.) What trends or patterns do you notice that indicate strengths and/or areas of need for the class as a whole, subgroups or individuals? What will be your next steps based on this analysis?

• The “YES/NO” cards described earlier were also used as a formative assessment technique in this lesson. As is described below, most students’ problems did not stem from a lack of understanding of phase shifts—100 percent of students were able to identify the amount by which a function should be shifted and whether the graph moved to the left or right. Problems arose, however, when they were asked to provide the new values for the graph—for example, six out of 16 students incorrectly responded that the graph of cos(x + π) would start at 0 (it starts at -1). Overall, no more than 12 out of 16 students correctly responded to each question; three of the four students mentioned earlier who scored below 50 percent on the most recent “speed test”— Students 3, 5, and 10—failed to respond correctly to any questions. As is described below, I attempted to build remediation into the lesson rather than re-covering the values directly; I also suggested these students could work with me during FLEX time to improve their understanding of the material.

Reflection What instructional and/or assessment activities were effective? Why? What instructional and/or assessment activities were less effective? Why? What changes would you make to your instructional and/or assessment procedures that would improve student learning? How did you “think on your feet” to make instructional decisions while you were teaching the lesson? What worked or did not work about these decisions? How did the oral or written feedback you gave students address their needs in relation to meeting the lesson objective? What multicultural considerations emerged during this lesson that you had not anticipated? How would you address those in the future? What insights have you gained from teaching this lesson?

• It became apparent as the lesson progressed that many students—those who struggled on the most recent “speed test”— were going to encounter difficulty with this topic due to insufficient prerequisite knowledge. As I had expected based on the pre-test, most students’ problems stemmed not with period changes, but with phase shifts. Students had no problem identifying these shifts when given a function, but many ran into difficulty when it became necessary to graph these shifts. The most frequent problem observed was that students did not know where to start the graph, which is a direct consequence of not knowing the relevant values. Although I did not include any direct remediation of this topic in the lesson, I attempted to integrate it as much as possible, such as by having students compose a table of values to illustrate a horizontal shift. In addition, due to this necessary remediation, the assessments were changed slightly; students were instructed to complete the first handout for homework, and examples demonstrated for the class, which were originally taken from the inclass handout, were instead taken from a worksheet used during a previous semester. This adjustment was necessary to ensure all students would be able to complete the problems. • Both my mentor and I reiterated throughout the lesson that at this point in the semester, all students are expected to have a thorough knowledge of quadrantal trigonometric values. We were careful to do this without singling out any students; if one student struggled to recall a value, the class as a whole was reminded of the importance of solidifying this knowledge base. • Like the first class, students enjoyed the examples given in the PowerPoint to illustrate the concepts of frequency and period. Using two items from school with which all students are familiar proved to be successful at engaging all learners, as intended.

V. InTASC or COE Standard and Rationale - How does this lesson represent my progress toward gaining mastery of the skills of an effective teacher? This lesson is a good example of InTASC or COE ____ because . . . â&#x20AC;˘ InTASC 2 (Learning Differences): Winters Mill has the highest percentage of FARMS students in Carroll County. Any lesson plan must remain cognizant of this fact and not include any components that may exclude FARMS students, such as determining the circumference of a backyard swimming pool or viewing planets through a telescope. Certainly, while examples of these sorts of problems abound in math classes, it would be patently unfair to FARMS students to include them; this would also likely result in adverse learning outcomes for these students. All examples used in this lesson have been carefully selected to ensure they represent experiences that all students, regardless of socioeconomic status, will have had the chance to experience, such as attending a WMHS volleyball match. (This is a theme throughout the entire unit, but it is particularly applicable to this lesson given the significant number of applications.) By giving all students examples to which they can easily relate, this lesson will allow them to fully realize the power of trigonometric functions and achieve to the best of their abilities. â&#x20AC;˘ InTASC 6 (Assessment to Prove and Improve Student Learning): The primary format of any assessment in a mathematics class is typically a series of problems to solve or graph. While this does most accurately reflect the core material, it fosters an environment in which mathematics is taught in a vacuum and applications are virtually ignored. While the assessments for this lesson do include functions to be graphed and equations to be solved, there are also a handful of word problems in which students must model a given situation using trigonometric functions and the techniques discussed in class. The formative assessments also include a component in which students will be given a graph and must identify the corresponding equationâ&#x20AC;&#x201D;the exact opposite of how such problems are typically structured. The wide variety of assessment techniques is designed to ensure students are both fully aware of the applications of the material and have acquired sufficient breadth in their study of it.

1/20/14&

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Unit Trigonometric Functions Grade Class Size 12 16 School Winters Mill High School

Topic Combinations of Translations Time 1:00-2:20 p.m. Intern Ben Cohen

Other: Maryland Content Standards

2.1.2 The student will identify the domain, range, the rule or other essential characteristic of a function. 2.2.3 The student will perform translations, reflections, and dilations on functions. Big Idea or Essential Question This lesson continues our study of translations of the sine and cosine functions; today, we will focus on combinations of horizontal and vertical translations. Students will learn techniques for graphing these combinations of translations—including the order that must be followed when graphing—as well as the applications of these combinations of translations, such as sound waves (the examples of guitar solos from the first class will be revisited, but now students will be able to analyze both why some waves are taller than others and why some are more “bunched up”).

Alignment with Summative Assessment Question 7 on the post-test asks students to graph a sine function given both horizontal and vertical translations. Lesson Objective Students will be able to graph combinations transformations of sine and cosine functions and will be able to identify their amplitude, vertical shift, period, and phase shift. Formative Assessment Following the lecture portion of the lesson, students will complete an eight-question problem set in class. Students will also complete an additional 12 questions for homework. All questions on the classwork and six questions on the homework will ask students to graph sine and cosine functions that have been transformed both horizontally and vertically; six additional questions on the homework will ask students to identify the amplitude, vertical shift, period, and phase shift of a given sine or cosine function.

►►►►►

(What prior knowledge of learners are you using to plan this lesson? DATA

Instructional Decisions Based on Knowledge of Learners RESPONSE TO DATA (required)

Formal data - Pretest, formative assessment, checklists, etc.

• Only one student (Student 10) was able to graph a function with a combination of translations on the pre-test.

• Although students should be familiar with this from their Algebra II work, it is likely that much of their difficulties stem from having never seen combinations of translations in this context—much of their prior work with translations has involved parabolas and graphs of the absolute value function. All of the material in this lesson will be introduced as though students had never seen it previously; if it appears that they are catching on more quickly than expected, the lesson will be adjusted accordingly. • This shows that students are aware of what needs to be done to translate a graph; their problems occur when it comes to determining how to implement the translations in the actual graph, with much of their difficulties traceable to a lack of adequate scaffolding (i.e., quadrantal trigonometric values). In turn, this suggests that when it comes to combinations of translations, students will be able to easily grasp this in the abstract—for example, “move the graph up three and to the left π”—but may need some help with the graphing portion, particularly to ensure they are implementing the translations in the proper order.

• During the previous lesson, 100 percent of students responded correctly when asked about the magnitude and direction of a vertical or horizontal shift.

Informal data - Observations of students, reflections from previous lesson(s), anecdotal records, etc.

• When demonstrating the graph of functions at the board or using the ActivInspire tablet, my back is to the students; in addition, the ActivInspire tablet is not large enough to graph more than one function at a time.

Individual or Small Group Needs Specific to this Lesson

• For this lesson, rather than using the ActivInspire, I will prepare a series of transparencies and display them using the overhead projector. In addition to allowing me to look at the class while I am putting up a graph—and thus see if any students look particularly puzzled—this will allow me to display multiple graphs at the same time and highlight their similarities and differences, as well as instantly change colors so I can compare a parent function to its translated graph.

►►►►►

(Ex. IEP/504 accommodations, ELL, social concerns, gifted/talented, multicultural/equity measures, etc.)

Differentiated Practices Specific to this Lesson (Instruction and/or Assessment)

• Student 4 has a 504 plan due to hearing loss.

• He is seated in the front of the classroom during all class meetings (students’ seats were assigned at the start of the academic year). I will need to speak slightly louder than usual during this class, both during lecture and as I am circulating the classroom as students are working individually. In addition, I will need to repeat questions posed by students to the whole class so he can understand (e.g., “Allison just asked…”). When the examples of music are re-visited from the first class, I will make sure to describe each clip (e.g., “soft and fastpaced”) to ensure he understands the corresponding period and amplitude changes.

• During the last lesson, it became necessary to devote time to remediation of trigonometric values, which it was expected students would master by this point in the semester. The need was particularly acute for Students 3, 5, and 10, all of whom scored below 50 percent on the most recent “speed test.”

• I will again work remediation into this lesson—not directly, but by asking students to provide quadrantal values as I am constructing the graph of a function. In addition, I will ask all students to use the “YES/NO” cards to explain whether they knew all values that were provided among a given period; this will allow me to monitor these students’ achievement without embarrassing them by asking them specifically.

III. Instructional Procedures – What instructional strategies will I use to ensure that every child is a successful learner? Instructional Materials/Resources A handout will be used in class that contains problems for the students to work out, working individually; a second handout will be prepared for homework. Both handouts will include spaces for students to graph the functions, rather than using a separate handout with a graph on it. Students will also be given “YES/NO” cards like in the previous lesson; these will be used to measure their understanding of the lesson content. This lesson will also utilize the TI Nspire calculators provided in the classroom and a series of overhead transparencies to graph functions (detailed below).

Technology Integration All students will use the TI Nspire calculators in the classroom to model the combinations of translations covered in this lesson. This will only be done to aid students’ understanding of the concepts; students will be required to use pencil-and-paper methods on the formative assessments. In addition, the instructor will prepare a series of overhead transparencies to use when graphing the functions. While previous lessons had used the ActivInspire tablet to annotate the computer screen, given the complexity of the translations in this lesson, the decision was made to use transparencies instead given the ease with which colors can be changed as well as the ability to display multiple problems simultaneously (the ActivInspire tablet is not large enough to allow for this). Cross-curricular Connections This lesson builds on the connections introduced in the first lesson. As part of that lesson, students were shown a series of sound waves based on guitar solos and asked to compare the music they heard to the sound waves they were shown—for example, the first clip was soft and fast, yielding a sound wave with a low amplitude but high frequency. This lesson will allow them to revisit these examples, only now they will have the tools to come up with a basic model for the sound wave—in the example above, they would know the sound wave had a shorter period and lower amplitude than the following two clips.

Management Considerations (Procedures, Transitions, Materials) and Student Engagement (required) • Rather than wasting valuable class time, students will be asked to come to the front of the room to get their calculators when they walk into the room prior to the start of class. This both saves time during class and also eliminates any concerns about a temporary break in the instructional time; it also ensures students will have access to the calculators for the duration of the class, in case it becomes necessary to adjust the lesson plan “on the fly” and give them extra time to practice graphing functions with the calculators. In addition, classwork will be passed out at the start of class, as students are working on the warm-up, to ensure instructional time is not wasted while distributing this material. • Because of the built-in remediation, this lesson is slightly more lecture-heavy than the previous day’s; as such, care must be taken to ensure opportunities for student engagement are plentiful. The warm-up includes follow-up questions to engage students; in addition, following this, six students will be asked to come to the board and demonstrate problems from the previous day’s homework. Also, during the individual-work portions of the lesson, the instructor will circulate to actively engage students and ensure they are staying on task. The instructor will attempt to make the remediation as interactive as possible, such as by surveying the class to both provide trigonometric values and explain how the graph of these values would change following the application of the transformations. • The TI Nspire calculators used during this lesson are the property of the school and remain in the classroom at all times; students are not permitted to bring them home for work or practice. In addition, given the high percentage of FARMS students at the school, all students cannot be expected to own a graphing calculator of their own. As such, the instructor must be prepared to devote extra time to showing students tips and tricks for graphing these functions on the calculator, since unlike a TI-83, they are still learning how to use these calculators.

Instructional Sequence Planned Beginning • • •

Approximate Time 10 minutes

Warm-up Motivation Bridge

Procedure • For their warm-up activity, students will be given two functions— one a sine function that has been translated vertically, one a cosine function that has been translated horizontally—and will be asked to find the amplitude and vertical shift of the former and the period and phase shift of the latter function. As a follow-up question, the responding student will be asked to describe how these translations will affect the graph of the function. • This warm-up will ensure that students have sufficient knowledge of vertical and horizontal translations to begin today’s lesson: combining them in a single function.

30 minutes

• As a lead-in, begin by calling six students to the board to put up graphs of functions from the previous night’s homework. Each problem will be a sine or cosine function that has been transformed horizontally or vertically—but not both. Once all students have completed putting up the problems, go over each problem in order, either making sure all students understand why it is correct or noting and correcting any errors that were made in the drawing. • Once this is complete, begin with the first problem (a phase shift of the cosine function) and ask students to think about how the graph would differ if the graph were also stretched vertically. Solicit answers from students—with justifications—and then

demonstrate how this would affect the graph. • Using the TI Nspire emulator on the instructor’s computer, graph the original phase-shifted cosine function and then add the vertical stretch to the graph. Ensure students understand the reasoning behind the shape of the graph of the function. Encourage them to follow along on their calculators as well, or even to graph different combinations of translations of the parent function to ensure they fully understand. • Follow the same procedures with the remaining five functions on the board to demonstrate additional combinations of translations. Enrichment or Remediation

35 minutes

(As appropriate to lesson)

• Explain to students why the graphs appear the way they do, using a table of values to demonstrate the effect of each translation individually followed by their combination. Ask for student input to create the initial values as a means of integrating the necessary remediation into the lesson. • Introduce students to the required steps for graphing a combination of translations: identifying the period (to allow for proper labeling of the axes), determining whether there is any phase shift, and then applying any vertical translations. Remind them to follow order of operations when examining a combination of translations. • Begin working on a handout with eight problems on it (passed out at the beginning of class); all will ask students to graph a combination of translations of the sine or cosine functions. Tentatively, the instructor will lead the class through the first four examples and the students will complete the remaining four on their own; however, the number of instructor-led examples may be adjusted up or down based on student feedback and needs. The instructor will utilize transparencies and the overhead projector to graph these functions. • Circulate as students are working to answer any questions and informally assess students’ understanding of the material.

Assessment/ Evaluation

Ongoing

• Students will be informally assessed throughout the lesson. During the lecture component, the instructor will look around the room to see if any students are visibly having difficulty understanding. As students are working individually on their worksheets, the instructor will circulate to answer questions and ensure all students comprehend the material and are remaining on topic. • Take time throughout the lesson to ask if any students have questions about the material. • Additional formative assessment: For homework, students will complete a 12-question problem set similar to the one given in class. Six questions will ask them to graph a function given a combination of translations, while the rest will ask them to identify the amplitude, vertical shift, period, and phase shift of a given function.

Planned Ending (Closure) • •

5 minutes

Summary Homework

• Briefly summarize and restate the information that was covered during the lesson. • Pass out homework (handout with 12 problems).

• Five out of six students called to the board to put up homework problems successfully graphed the function. As I intentionally chose a heterogeneous sample with respect to ability levels—two higher achieving students (Students 11 and 12), two middle-of-thepack students (Students 6 and 7), and two students who had been having difficulty with the material (Students 4 and 13; the latter was the only student to provide an incorrect graph)—this can reasonably be interpreted to mean the class understands how to shift a graph horizontally or vertically, and no further remediation is necessary. There are likely still some students who will struggle simply because of their lack of command of trigonometric values, as has been described earlier, but I have determined, in conjunction with my mentor, that at this point it would be inappropriate to build in any remediation besides what is already integrated in the lesson, given that it is almost halfway through the marking period and well past the point at which students were expected to fully master this knowledge. • Three students (Students 1, 2, and 5) struggled with combinations of translations due to an inability to manipulate fractions. As this is knowledge that is expected to be mastered well before 12th grade, it would have been inappropriate to do a whole-class remediation; instead, I worked with these students individually during the individual-work portion of the lesson. This is likely not indicative of any particular failing on the students’ part, but rather the product of their reliance on calculators to manipulate fractions; a quick refresher appeared to jog these students’ memories on how to perform these operations by hand. • Two students (Students 15 and 7) encountered difficulty with this for a different reason: They interpreted sin(-x) and –sin(x) as identical (and did the same with cos(-x) and –cos(x)). It is easy to see where the error came from—they are used to factoring out the negative sign, although in the case of some functions this cannot be done—and this was easily correctible and does not appear to be indicative of any underlying difficulties with the material. • Seven out of 16 students failed to label the x-axis on problem 5 on the classwork. This is likely because this problem involved only a vertical translation, rather than a period change or phase shift, and thus the normal x-scale is in effect; however, I will need to remind the class during the next lesson that the x-axis must always be labeled, rather than simply leaving the reader to assume the period is 2π unless otherwise indicated.

• Certainly, I was not expecting to have to devote part of the lesson to remediation of adding and subtracting fractions. Although it was distressing to see three students unable to perform this operation, as was described earlier, it appears this was primarily a byproduct of their reliance on calculators to perform these operations, since these students appeared to quickly grasp the methods once I demonstrated for them. Interestingly, these students did not come right out and tell me that they were having problems manipulating the fractions—only after repeated questioning as I tried to lead them to the solution did they admit this was giving them trouble. This seems to suggest that the students were aware that this is something they should have been able to easily perform. While I felt it would not be appropriate to do a whole-class remediation of fraction operations, this did suggest that in future lessons, it might be advisable when going through the table of values to include a fraction-manipulation component just to reactive students’ knowledge of how to perform these operations by hand. • At the suggestion of my mentor, I tried using a slightly different method than what I had planned to teach combinations of translations. Rather than forming a table of values and then graphing all of the points at once—the method I had used in high school—he encouraged me to use the “ ’X’ marks the spot” method, which consists of placing an “X” where the first point on the function would normally be based on the parent function and period, and then applying the vertical translations and phase shift to determine where the first point should actually go. This proved to be a highly effective method, and I will certainly use it when I teach this unit in the future. • Some students remarked that it would have been helpful to have the graphs of the functions we went over together already on their calculators to facilitate further exploration—this way, the students would not have had to take the time to enter the parent function and adjust the viewing window, but could have simply gone right ahead and begun translating the function. This is a very valid point, and so for the remaining lessons, I will load all such graphs onto students’ calculators in the morning prior to the start of class.

V. InTASC or COE Standard and Rationale - How does this lesson represent my progress toward gaining mastery of the skills of an effective teacher? This lesson is a good example of InTASC or COE ____ because . . . • InTASC 3 (Learning Environments): Despite this being a lecture-based lesson (due to needed remediation), many steps were taken to ensure active student engagement and thought throughout the lesson. For example, when going over the warm-up problem, respondents were asked to determine how the translations they were identifying would affect the graph of the function; this was designed to get them thinking about combining translations. In addition, when students were called to the board to put up homework problems, rather than simply announcing each was correct or incorrect, I surveyed the students to see if anyone had any questions/objections to each graph or believed it to be correct; this proved to be a valuable tool, as students were not only able to correctly identify the errors in the lone incorrect graph, but engaged in discussions with each other about the graphs and why they looked the way they did. I also went to great lengths to not marginalize the students who were having difficulty with fraction operations; while this is prerequisite content knowledge that they should certainly have mastered by now, I did not want to jump over them for not knowing this, for this would serve only to create a hostile learning environment and an adversarial relationship between us. Instead, I patiently worked through their difficulties with them; they quickly realized what they were doing wrong and were able to rectify the issue. While it would be tempting to state that I should have also reminded them that they should know this by now, the students were likely well aware of this; my job is to help them work through their difficulties with the material, not to sharply criticize them for these difficulties, whatever their cause may be. • InTASC 10 (Leadership and Collaboration): Initially, I had planned to introduce combinations of transformations using the same method I did when I took pre-calculus: making a table of values to determine the end result of the translation on each point on the graph over a two-period interval. Though tedious, this method worked for me and ensured both a thorough understanding and accurate graph. However, my mentor suggested that I use a different method, which he referred to as “ ‘X’ marks the spot.” The idea behind this method is to run through the procedure in reverse: Start by plotting the starting point of the parent graph, translate it horizontally and vertically, and then use the known pattern of the parent graph to fill in the remaining points. Once this is done, there is ample opportunity to discuss why students have observed these particular patterns. We had to run through a few examples together prior to class since this was an entirely new method to me, but it proved to be extremely useful for getting the students to grasp some of the hardest material in their class, and I will be sure to use the same method when I am teaching this unit in future classes. I have of course been in contact with my mentor throughout this semester regarding the best approach to take with the material, but this method in particular stands out to me.

Classroom Management Matrix

Intern: Ben Cohen

School: Winters Mill High School

Date: 10/10/13

Planning for Effective Lesson Management Objective: Graph combinations of transformations of sine and cosine functions. Lesson Sequence

Beginning the Lesson (Including transitions)

Teaching the Lesson

Ending the Lesson

Classroom Routines and Procedures (Students)

Teacher Preparation and Organization

Physical Arrangement and Use of Space

Student Engagement

Behavior Management (Whole Group and Individual)

Homework will be collected and classwork passed out while students are working on the warm-up. Write on the board that they will not need their textbooks today. They are aware that they are expected to retrieve their calculators unless otherwise notified at the start of class.

All handouts and transparencies will be prepared prior to the lesson. The overhead projector will be in place and ready for use by the start of class.

Students were assigned seats at the start of the semester; these will remain unchanged. Only the front whiteboard will be used (it is large enough to fit six problems).

Use the football to give students a chance to volunteer answers to the warm-up and come write homework problems up on the board.

There are no significant behaviorial issues in this class; just the same, circulate during this portion of the lesson to ensure students are remaining on task. They are less likely to stray off task if I am physically present.

Pay particular attention to Student 4 during the lecture portions of the lesson— surreptitiously ensure he can hear what both the other students and I are saying. Make sure students know they should ask questions as they arise and not simply when I ask if there are any questions. Leave 5-10 minutes at the end of class to go over the classwork and take questions from students. They are used to this routine from previous units and it has proven to be successful. Brief them on the plan for tomorrow (cosecant and secant graphs).

All of the problems that will be demonstrated in this lesson will be graphed in advanced so I do not have to waste time thinking of how to do it; students will be taken through step by step, but I should not be thinking out loud as I am doing this. Homework will be passed out as students are checking their answers to the classwork—do not pass it out earlier; otherwise, students may work on it instead of the classwork.

Ensure the overhead projector is placed in such a manner that it does not inhibit students’ ability to see the coordinate plane at the front of the room, nor my ability to circulate between rows.

This is a more lectureheavy lesson due to remediation; ample opportunities for student engagement are built in, such as surveying students to provide trigonometric values of angles. Use the football to ensure active participation! Again using the football, ask a few questions summarizing the information that was covered in the lesson. Ensure all students have had at least one opportunity to participate.

Again, since this is a more lecture-heavy lesson, behavior management is more crucial. Ensure sufficient student engagement throughout the lesson and circulate as necessary to assist students and ensure they are on task. Students in this class generally do not have a problem remaining seated until the bell; just the same, do not say or do anything (e.g., “see you tomorrow!”) to suggest they can start lining up.

Again, ensure both the students and I have an unimpeded path between desks. Pass out assignments to each student individually to ensure they remain on task at the end of the lesson.

Unit Trigonometric Functions Grade Class Size 12 16 School Winters Mill High School

Topic Cosecant & Secant Functions Time 1:00-2:20 p.m. Intern Ben Cohen

Other: Maryland Content Standards

2.2.3 The student will perform translations, reflections, and dilations on functions. Big Idea or Essential Question This lesson moves beyond the basic sine and cosine functions into a study of their respective inverses, the cosecant and secant functions. These functions share many common traits but also have a number of pronounced differences; by developing this knowledge, students will gain a deeper understanding of the relationship between the sine and cosine functions and their inverses, as well as the role of trigonometric functions throughout historyâ&#x20AC;&#x201D;the cosecant and secant functions were used in navigation and cartography as early as 1645, and this historical role will be a focal point of the lesson.

Alignment with Summative Assessment Question 6 on the post-test asks students to graph a secant function, with both horizontal and vertical translations. Lesson Objective Students will be able to graph the cosecant and secant functions, including horizontal and vertical translations and combinations of these translations. Formative Assessment Following the lecture portion of the lesson, students will complete a 12-question problem set in class. Students will also complete an additional 14 questions for homework. All questions on the classwork and eight questions on the homework will ask students to graph cosecant and secant functions, including both vertical and horizontal transformations; the remaining six questions on the homework will ask students to identify the amplitude (if any), vertical shift, period, and phase shift of a given function.

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(What prior knowledge of learners are you using to plan this lesson? DATA

Instructional Decisions Based on Knowledge of Learners RESPONSE TO DATA (required)

Formal data - Pretest, formative assessment, checklists, etc.

• No students were able to successfully graph a secant function on the pretest.

• This is not surprising, since students’ only prior exposure to the cosecant and secant functions has come when they were exploring values and identities concerning the functions, not graphs. Just the same, this confirms that this is entirely new material for the students, and I will need to treat it as such, including devoting time to showing them the tricks to using parent functions (sine and cosine, respectively) to graph cosecant and secant.

• This was used in the planning of the first lesson, but it bears repeating here. Given the structure of these functions, it is crucial that students have a solid grasp of the values of sine, cosine, cosecant, and secant for quadrantal angles. Unfortunately, question-by-question statistics are not available for the last “speed test,” but given the structure of the test, a score below 50 percent indicates that the student lacks knowledge of at least two of these functions. As was done previously, I will build remediation into the lessons, such as by asking students to provide the values of sine and cosecant or cosine and secant as I show the class how to graph these functions.

Informal data - Observations of students, reflections from previous lesson(s), anecdotal records, etc.

• Students are already familiar with the concept of asymptotes from their Algebra II work.

Individual or Small Group Needs Specific to this Lesson

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(Ex. IEP/504 accommodations, ELL, social concerns, gifted/talented, multicultural/equity measures, etc.)

• Student 4 has a 504 plan due to hearing loss.

• The concepts of asymptotes and convergence to infinity play a central role in the graphing of the cosecant and secant functions. Although I will be prepared to introduce a remediation component if necessary, I will plan on simply providing an overview of these functions’ asymptotic behavior at given points, rather than defining this concept upon its introduction.

Differentiated Practices Specific to this Lesson (Instruction and/or Assessment)

• He is seated in the front of the classroom during all class meetings (students’ seats were assigned at the start of the academic year). I will need to

speak slightly louder than usual during this class, both during lecture and as I am circulating the classroom as students are working individually. In addition, I will need to repeat questions posed by students to the whole class so he can understand (e.g., “Allison just asked…”). • Although part of this lesson is devoted to calculator techniques to graph these functions, I must remain cognizant of the high percentage of FARMS students.

• Some students in the class own a TI-83 or TI-84 graphing calculator; although this will allow them to experiment with translations of the cosecant and secant functions on their own, there are some challenges they must be made aware of prior to doing this (such as changing the graphing mode on the calculator). However, as I am at the school with the highest percentage of FARMS students in Carroll County, it is highly likely that other students in the class cannot afford a graphing calculator of their own. Thus, in addition to providing time in class to experiment with translations on a TI Nspire calculator, I must broach this subject carefully. Rather than asking which students in the class own a TI-83 or TI-84—even if I were to use the “YES/NO” cards, this could create an uncomfortable situation for some FARMS students—I will simply preface this portion of the lesson with “If you have a TI-83 or TI-84…” to avoid singling out the more fortunate students.

III. Instructional Procedures – What instructional strategies will I use to ensure that every child is a successful learner? Instructional Materials/Resources A handout will be used in class that contains problems for the students to work out, working individually; a second handout will be prepared for homework. Both handouts will include spaces for students to graph the functions, rather than using a separate handout with a graph on it. Students will also be given “YES/NO” cards like in the previous two lessons; these will be used to measure their understanding of the lesson content. This lesson will also utilize the TI Nspire calculators provided in the classroom and a series of overhead transparencies to graph functions (detailed below).

Technology Integration All students will use the TI Nspire calculators in the classroom to model the cosecant and secant functions and their horizontal transformations (graphs of all functions that will be demonstrated for the class will be loaded onto all students’ calculators prior to class). This will only be done to aid students’ understanding of the concepts; students will be required to use pencil-and-paper methods on the formative assessments. A portion of the lesson will also be devoted to techniques for graphing these functions on TI-83 and TI-84 calculators for those students who utilize these devices. In addition, the instructor will prepare a series of overhead transparencies to use when graphing the functions, as was done when graphing combinations of translations. Cross-curricular Connections This lesson includes a connection to students’ studies of history and geography. Students will be informed that the cosecant and secant functions played a key role during 17th-century maritime navigation and cartography; they will also have the opportunity to brainstorm potential uses for the

functions in these fields. While there are other applications of these functions, they require knowledge of mathematics well beyond what high-school students can reasonably be expected to know; as such, examples will be limited to the areas of study listed above, which will afford all students the opportunity for involvement and understanding. Management Considerations (Procedures, Transitions, Materials) and Student Engagement (required) • Rather than wasting valuable class time, students will be asked to come to the front of the room to get their calculators when they walk into the room prior to the start of class. This both saves time during class and also eliminates any concerns about a temporary break in the instructional time; it also ensures students will have access to the calculators for the duration of the class, in case it becomes necessary to adjust the lesson plan “on the fly” and give them extra time to practice graphing functions with the calculators. In addition, classwork will be passed out at the start of class, as students are working on the warm-up, to ensure instructional time is not wasted while distributing this material. • Opportunities for student engagement are included throughout the lesson. Students will be asked to volunteer answers to the warm-up questions, the differences between shifted graphs, and values of cosecant and secant for quadrantal angles. They will also be asked to brainstorm scenarios in which the cosecant and secant functions may be useful, given their role in cartography and navigation, which will be described in the lesson. In addition, during the individual-work portions of the lesson, the instructor will circulate to actively engage students and ensure they are staying on task. • The TI Nspire calculators used during this lesson are the property of the school and remain in the classroom at all times; students are not permitted to bring them home for work or practice. In addition, given the high percentage of FARMS students at the school, all students cannot be expected to own a graphing calculator of their own. As such, the instructor must be prepared to devote extra time to showing students tips and tricks for graphing these functions on the calculator, since unlike a TI-83, they are still learning how to use these calculators.

Instructional Sequence Planned Beginning • • •

Approximate Time 10 minutes

Warm-up Motivation Bridge

Procedure • For their warm-up activity, students will answer the following true/false questions using their “YES/NO” cards: 1) Starting at the origin, the graph of y = − cos(x) goes up, up, down, down. 2) The amplitude of y = 3sin(x + π ) is π .

y = cos(2(x − π )) has a period of π . π π 4) The graph of y = sin(x − ) is shifted units to the left. 2 2 1 π 5) The function y = sin( x − ) has a horizontal compression 2 4 1 of . 2 3) The function

• Up to this point, all of our work has concentrated on sine and cosine functions. What about the graphs of their reciprocal functions—cosecant and secant? Development of the New Learning

35 minutes

• Begin by projecting a table of values for the sine function. Ask students to examine the value of the cosecant of x at x = π/2:

csc(x) = 1. What about at x = π? Since sin(π) = 0, csc(π) is undefined; the same situation occurs at x = 0. But what about as we get closer and closer to x = 0 and x = π? Sin(x) continues to get smaller and smaller, so 1/sin(x) (i.e., csc(x)) will get larger and larger.

(Procedure will vary with the instructional model used. Ex. 5E lesson plan, scientific inquiry, teacher/student modeling, cooperative learning, before, during and after strategic reading, etc. Focus on active student engagement.)

• Using the TI Nspire emulator on the instructor’s computer, graph the sine function and then follow by graphing the cosecant function. Ensure students understand the reasoning behind the graph of the function and the locations of the vertical asymptotes, as well as how to use the sine function as a reference. • Follow the same procedures to show students how to graph the secant function using the cosine function as a reference. • Pose the question to students: Are the cosecant and secant functions periodic functions? Survey the class as to its thoughts, then ask for the opinion of someone who answered “yes” and someone who answered “no.” If all students answer “yes” or “no,” see if there are any differences of opinion as to why this is the case. • Explain that although they are different in appearance than the sine and cosine functions, cosecant and secant are in fact periodic functions—they, too, repeat over the interval of 2π. • Now ask students if the cosecant and secant functions have an amplitude. Once again, survey the class and have one or more students explain their answers. This time, explain that these functions do not have an amplitude, since they extend infinitely and thus the distance between the maximum and minimum values cannot be determined.

Enrichment or Remediation (As appropriate to lesson)

30 minutes

• Like sine and cosine functions, the cosecant and secant functions can be transformed vertically and horizontally. Run through two examples of transformations (one cosecant, one secant) to show students the technique for graphing these transformed functions: graph the transformed sine or cosine function as a reference, and use this function to graph the cosecant or secant function. • Calculator note: Remind students that there is no way to select “csc” or “sec” on a TI-83 or TI-84, unlike on the Nspires; they must graph 1/sin(x) or 1/cos(x). In addition, on a TI-83, when graphing in “Connected” mode, vertical lines will appear on the graph to illustrate the asymptotes. To remove these lines, students can switch to “Dotted” mode. • Introduce students to problems that can be modeled using the cosecant and secant functions—in this case, those in the fields of cartography and navigation. While the mathematics required to solve such problems is far beyond the students’ level, ensure they are aware of the uses and applications of these functions. • Distribute a worksheet with 12 functions on it (vertical and horizontal transformations of cosecant and secant functions). The

class will do the first four examples together, after which students will complete the rest of the worksheet individually. All 12 questions will ask students to draw a transformed cosecant or secant graph. • Circulate as students are working to answer any questions and informally assess students’ understanding of the material. Assessment/ Evaluation

Ongoing

• Students will be informally assessed throughout the lesson. During the lecture component, the instructor will look around the room to see if any students are visibly having difficulty understanding. As students are working individually on their worksheets, the instructor will circulate to answer questions and ensure all students comprehend the material and are remaining on topic. • Take time throughout the lesson to ask if any students have questions about the material. • Additional formative assessment: For homework, students will complete a 14-question problem set similar to the one given in class. Eight questions will ask them to determine the graph of a transformed/translated function, while the rest will ask them to identify the amplitude (if any), vertical shift, period, and phase shift of a given function (sine, cosine, cosecant, or secant).

Planned Ending (Closure) • •

5 minutes

Summary Homework

• Briefly summarize and restate the information that was covered during the lesson. • Pass out homework (handout with 14 problems).

• The following numbers of students correctly answered the warmup questions: • Question 1 (true): 11/16 (69%) • Question 2 (false): 14/16 (88%) • Question 3 (true): 14/16 (88%) • Question 4 (false): 16/16 (100%) • Question 5 (false): 13/16 (81%) This suggests that the bulk of the class has mastered the concepts from the previous three lessons, which should in turn be a harbinger of success for the remaining two (although some students can use some remediation on vertical reflections of functions, the subject of the first warm-up question). The purpose of this warm-up was to allow me to determine whether any remediation was needed at the start of the lesson; other than a quick reminder about function negation, it appears none was necessary. • Eleven of 16 students (69%) correctly graphed at least four out of the six independent-work problems on the classwork; the most

common errors noticed were not substantial, but rather minor mistakes like shifting the graph up two places instead of one. Interestingly, three out of 16 students, on at least one problem, graphed the parent function properly but did not add in the parabolas of the cosecant or secant function; as they graphed both functions correctly on other problems, it is very likely that this was simply a minor oversight. Reflection What instructional and/or assessment activities were effective? Why? What instructional and/or assessment activities were less effective? Why? What changes would you make to your instructional and/or assessment procedures that would improve student learning? How did you “think on your feet” to make instructional decisions while you were teaching the lesson? What worked or did not work about these decisions? How did the oral or written feedback you gave students address their needs in relation to meeting the lesson objective? What multicultural considerations emerged during this lesson that you had not anticipated? How would you address those in the future? What insights have you gained from teaching this lesson?

• Some students are still unaware of the definitions of cosecant (inverse sine) and secant (inverse cosine); again, at this point in the semester, this is knowledge that students need to have down pat. While it is possible that these students will be able to successfully graph the cosecant and secant functions merely by memorizing the appearances of their respective graphs, this is not a sustainable long-term strategy given that they will be asked to graph translations of these graphs. I attempted to work remediation into the lesson by reminding the class as a whole of the definitions of cosecant and secant; at this point in the semester, it would not be appropriate to devote a significant portion of a lesson to the derivation of these functions, which has been covered previously, to benefit an extreme minority of students. • Students initially complained about the number of problems they were given (12 in class and 14 more for homework). However, the best way to master the material in this unit is through repetition, and many students later remarked to me that they found the emphasis on practice quite beneficial. • One consideration I had not thought about previously came up when I was modeling the asymptotic behavior of these functions. The exact terminology I used was that the value of csc(x) “gets closer and closer to infinity as x approaches 0.” However, since this class is Trigonometry/Pre-Calculus, I wondered whether it would be appropriate to introduce the concept of limits and convergence, which play a central role in calculus. I decided that since this is a level 6 (standard) class composed entirely of seniors who are not on track to take calculus in the spring, it was best not to mention these. If, however, it were a level 8 (honors) class whose students were on the calculus track, I would certainly have introduced these concepts. I will keep this in mind the next time I teach this unit.

V. InTASC or COE Standard and Rationale - How does this lesson represent my progress toward gaining mastery of the skills of an effective teacher? This lesson is a good example of InTASC or COE ____ because . . . • InTASC 4 (Content Knowledge): Unquestionably, the material in this unit is fairly dry. This is particularly true for the cosecant and secant functions, since unlike their parent functions (sine and cosine), they cannot be used to model many real-world phenomena. In addition, at first glance these functions appear to have been derived in a vacuum. For these reasons, I felt it was extremely important to note the applications of these functions in the fields of maritime and navigation, and particularly that they had been used in these fields for close to 400 years. Although students lack the mathematical background to work out any such problems—which would require years of advanced study—it was crucial for them to learn that there do exist myriad applications of these functions. In addition, I have attempted throughout this unit to link new concepts to familiar ones—such as tying sine and cosine function graphs to sound waves from a guitar—and this certainly fulfills that aim. As I have remarked to students multiple times, this unit is likely the hardest in the Trigonometry/Pre-Calculus curriculum; by relating the content to concepts with which students are innately familiar, I hope to make it easier for students to understand both the content and how it affects their everyday lives. • InTASC 9 (Professional Learning and Ethical Practice): The warm-up activity for this lesson was specifically designed as a midpoint review of sorts to assess the effectiveness of my instruction; although I have been able to obtain informal feedback through conversations with students and my mentor, I wanted to have an additional data-based feedback mechanism (in addition to the formative assessments, which would not reflect any sudden insights or realizations by students following the lesson). In addition, this allowed me to determine how to structure this lesson and the following one, including whether any additional remediation of concepts would be necessary and whether students had acquired sufficient knowledge to move forward within the unit. The questions were also structured to assess the source of students’ difficulty, if any—for example, question 5 was designed to ensure students are aware that fractional coefficients on the variable rather than on the function will stretch, rather than compress, the graph. Teaching is a constantly evolving process full of ongoing self-evaluations, and the more feedback I can get from those I work with, the better; this was another means of ensuring I have sufficient data on the effectiveness of my educational practices to determine what works, what doesn’t, and how future lessons should be structured.

Unit Trigonometric Functions Grade Class Size 12 16 School Winters Mill High School

Topic Tangent & Cotangent Functions Time 1:00-2:20 p.m. Intern Ben Cohen

Other: Maryland Content Standards

2.1.2 The student will identify the domain, range, the rule or other essential characteristic of a function. 2.2.3 The student will perform translations, reflections, and dilations on functions. Big Idea or Essential Question This will introduce the final two trigonometric functions, concluding our study: the tangent and cotangent functions. These functions differ slightly from the previous four functions introduced—for example, they have a period of π rather than 2π—but also share many similarities. Once again, this will give students a deeper appreciation for the relationships between trigonometric functions and identities, particularly now that they will have studied all six functions. They will also explore the role these two functions play in such fields as maritime navigation and architecture.

Alignment with Summative Assessment Question 2 on the summative assessment asks students to determine the phase shift of a given tangent function; question 5 asks them to graph a cotangent function, with transformations. Questions 9 and 10 ask students to solve a problem involving simple harmonic motion using trigonometric functions. Lesson Objective Students will be able to graph the tangent and cotangent functions, including horizontal and vertical translations and combinations of these translations. They will also be able to use the trigonometric functions they have learned to solve problems involving simple harmonic motion. (Note: Simple harmonic motion was originally planned to be introduced during lesson 2, but due to the need for remediation that arose during the lesson, this topic was not able to be covered in the necessary detail. As such, it has been added to today’s lesson, since the tangent and cotangent functions require less instructional time than the other trigonometric functions.) Formative Assessment Following the lecture portion of the lesson, students will complete a 14-question problem set in class. Eight questions will ask students to graph tangent and cotangent functions, including horizontal transformations (vertical transformations are excluded due to the nature of the function, although these will be demonstrated for the class); four more will ask them to identify the period, amplitude, and phase shift of a given function. The final two questions are word problems involving simple harmonic motion.

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(What prior knowledge of learners are you using to plan this lesson? DATA

Instructional Decisions Based on Knowledge of Learners RESPONSE TO DATA (required)

Formal data - Pretest, formative assessment, checklists, etc.

• No students were able to graph a cotangent function on the pretest.

• Much like the secant function, this result is not surprising, since students’ only prior exposure to the tangent and cotangent functions involved values and identities involving the functions and not their graphical representations. As expected, I will need to treat this as entirely new material for students (with the exception of asymptotes, with which they are already familiar).

• Exactly half of students (eight out of 16) scored a 90% or higher (43/48) on the most recent “speed test” assessing their knowledge of trigonometric values. Four students scored below 50 percent on the test.

• As was stated in the previous lesson, questionby-question statistics are unfortunately not available for the “speed tests,” but given the structure of the test, a score below 50 percent indicates that the student lacks basic knowledge of the sine and cosine functions, which in turn determine the tangent and cotangent functions. Once again, I will build remediation into the lessons by asking students to provide angles at which tangent and cotangent are undefined and equal to 1, 0, and -1.

Informal data - Observations of students, reflections from previous lesson(s), anecdotal records, etc.

• The previous lesson confirmed that students are familiar with the concept of asymptotes from their Algebra II studies.

Individual or Small Group Needs Specific to this Lesson

• During the previous lesson, I was prepared to include a remediation component if it became necessary, but all students were familiar with the concept of vertical asymptotes and thus quickly grasped that portion of the lesson. As such, at the start of today’s lesson, I can simply ask students to provide values at which tangent and cotangent are undefined and then noting that the graph is asymptotic at these values, rather than having to explain the concept of asymptotic behavior first.

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(Ex. IEP/504 accommodations, ELL, social concerns, gifted/talented, multicultural/equity measures, etc.)

• Students 4 and 13 will miss this entire week while serving as counselors at CCPS Outdoor School (Hashawha).

Differentiated Practices Specific to this Lesson (Instruction and/or Assessment)

• Although knowing of their absences in advance helps, this will nonetheless be extremely disruptive. While the most pressing concern is ensuring they have time to make up the missed material, as their absence occurs in the middle of a unit it may also

be necessary to build in brief remediation of concepts discussed earlier in the unit. I must also make sure not to overwhelm these students, since they will have a week’s worth of work to make up for their other three classes as well; at the same time, they will be expected to make up all missed material and work in a reasonably expeditious manner. I will discuss with my mentor the appropriate steps to take to ensure these students’ continued success. • Although part of this lesson is devoted to calculator techniques to graph the tangent and cotangent functions, I must remain cognizant of the high percentage of FARMS students.

• Much like in the previous lesson, it is essential to strike a balance between providing students who own a graphing calculator with the tools they will need to be successful and not unnecessarily excluding those who do not own a graphing calculator. As I did in the previous lesson, rather than asking which students in the class own a TI-83 or TI-84, I will preface this portion of the lesson with “If you have a TI-83 or TI-84…” to avoid singling out any students.

III. Instructional Procedures – What instructional strategies will I use to ensure that every child is a successful learner? Instructional Materials/Resources A handout will be used in class that contains 14 problems for the students to work out, working individually. The handout will include spaces for students to graph the functions, rather than using a separate handout with a graph on it. Students will also be given “YES/NO” cards like in the previous two lessons; these will be used to measure their understanding of the lesson content. This lesson will also utilize the TI Nspire calculators provided in the classroom and a series of overhead transparencies to graph functions (detailed below).

Technology Integration All students will use the TI Nspire calculators in the classroom to model the tangent and cotangent functions and their vertical and horizontal translations. This will only be done to aid students’ understanding of the concepts; students will be required to use pencil-and-paper methods on the formative assessments. A portion of the lesson will also be devoted to techniques for graphing these functions on TI-83 and TI-84 calculators for those students who utilize these devices. In addition, as was done for the previous two lessons, the instructor will prepare a series of overhead transparencies to use when graphing the functions. Cross-curricular Connections As with the study of the cosecant and secant functions, many of the applications of the tangent and cotangent functions require knowledge of mathematics well beyond a high-school curriculum. However, there are still a number of relevant applications to which the students can be exposed. For example, they can use the tangent function to determine the height of a tree or flagpole based on the length of its shadow; as an accommodation for FARMS students, these applications will be explicitly based on objects easily observable on the Winters Mill grounds. An application of the cotangent function is determining the distance of a rotating beacon from a reference point—such as the distance of a ship from the shore based on the beacon of a lighthouse. To add an additional cross-curricular connection, word problems of this sort will reference the Sambro Island Lighthouse in Halifax, Nova Scotia, the oldest lighthouse in

North America, and 18th-century sailors making use of its light beacon. Problems involving simple harmonic motion will similarly provide a connection to fields such as physics (motion of a spring), music (sound waves), and sports (height of a bouncing ball), utilizing objects that will be familiar to all students. Management Considerations (Procedures, Transitions, Materials) and Student Engagement (required) • Rather than wasting valuable class time, students will be asked to come to the front of the room to get their calculators when they walk into the room prior to the start of class. This both saves time during class and also eliminates any concerns about a temporary break in the instructional time; it also ensures students will have access to the calculators for the duration of the class, in case it becomes necessary to adjust the lesson plan “on the fly” and give them extra time to practice graphing functions with the calculators. In addition, classwork will be passed out at the start of class, as students are working on the warm-up, to ensure instructional time is not wasted while distributing this material. • Opportunities for student engagement are included throughout the lesson. Students will be asked to volunteer answers to the warm-up questions, the differences between shifted graphs, and values of tangent and cotangent for quadrantal angles, as well as helping work out word problems that will be posed during the enrichment portion of the lesson. In addition, during the individual-work portions of the lesson, the instructor will circulate to actively engage students and ensure they are staying on task. • The TI Nspire calculators used during this lesson are the property of the school and remain in the classroom at all times; students are not permitted to bring them home for work or practice. In addition, given the high percentage of FARMS students at the school, all students cannot be expected to own a graphing calculator of their own. As such, the instructor must be prepared to devote extra time to showing students tips and tricks for graphing these functions on the calculator, since unlike a TI-83, they are still learning how to use these calculators.

Instructional Sequence Planned Beginning • • •

Approximate Time 10 minutes

Warm-up Motivation Bridge

Procedure • For their warm-up activity, students will be given two functions to graph—one a cosecant function and one a secant, both with combinations of translations. Once students have finished, a volunteer will be asked to graph each function on the board; the rest of the class will be asked to determine if each graph is correct or what needs to be fixed to make it correct. • We now conclude our study of the graphs of trigonometric functions with the tangent and cotangent functions.

30 minutes

• Ask students to provide the value of an angle at which the tangent is undefined (e.g., π/2); draw a vertical asymptote at this point on the coordinate plane. Now, as was done when demonstrating the graphs of the cosecant and secant functions, ask students what happens as the value of x approaches this value—it gets progressively larger in magnitude! Now, ask students what happens as we move counterclockwise along the unit circle—we will reach a value at which tan(x) = 1, then 0, then -1, and then gets progressively more negative; eventually we reach another asymptotic value. (All values pertain to quadrantal angles with which students are expected to be familiar.) • Graph four periods of the tangent function. Inform students that this is only to supplement their knowledge; normally they will only

be asked to graph it over one period (compared to the two periods used previously). • Repeat the same procedure for the values of the cotangent function; observe that it becomes increasingly negative as it approaches a vertical asymptote and increasingly positive immediately after one—the opposite of the tangent function. Once again, graph four periods of the function but inform students they will normally only be asked to graph one period. • Using the TI Nspire emulator on the instructor’s computer, project the graphs of the tangent and cotangent functions. • Pose the question to students: Are the tangent and cotangent functions periodic functions? Survey the class as to their thoughts, then ask for the opinion of someone who answered “yes” and someone who answered “no.” If all students answer “yes” or “no,” see if there are any differences of opinion as to why this is the case. • Explain that much like the cosecant and secant functions, although they are different in appearance than the sine and cosine functions, tangent and cotangent are in fact periodic functions. Observe also that these are different than the other periodic functions we have covered to this point; these functions have a period of π rather than 2π. • Use the same procedure to ask students if the amplitude can be determined for the tangent and cotangent functions. As with the cosecant and secant functions, since the graphs extend infinitely, the amplitude cannot be determined. Enrichment or Remediation (As appropriate to lesson)

20 minutes

• Like other trigonometric functions, the tangent and cotangent functions can be transformed vertically and horizontally. Run through two examples of transformations (one tangent, one cotangent) to show students the technique for graphing these transformed functions: find two adjacent asymptotes and divide the area between them into four equal parts, then evaluate the function at these key points. • Calculator note: Remind students that there is no way to select “cot” on a TI-83 or TI-84, unlike on the Nspires; they must graph 1/tan(x). In addition, on these calculators, when graphing in “Connected” mode, vertical lines will appear on the graph to illustrate the asymptotes. To remove these lines, students can switch to “Dotted” mode. • After students have become comfortable with the tangent and cotangent functions, introduce examples of how to use trigonometric functions to solve problems involving simple harmonic motion. Examples will include both solving for time in a given equation and determining an equation to model a given problem. • Distribute a worksheet with 14 problems on it (12 vertical and horizontal transformations of tangent and cotangent functions and

two word problems). The class will do the first three examples together, after which students will complete the rest of the worksheet individually. Six questions will ask students to draw a transformed tangent or cotangent graph; six more will ask students to identify the period and phase shift of a given function, and the remaining two will be word problems. • Circulate as students are working to answer any questions and informally assess students’ understanding of the material. Assessment/ Evaluation

Ongoing

• Students will be informally assessed throughout the lesson. During the lecture component, the instructor will look around the room to see if any students are visibly having difficulty understanding. As students are working individually on their worksheets, the instructor will circulate to answer questions and ensure all students comprehend the material and are remaining on topic. • The warm-up is also designed to serve as an informal assessment; the instructor will note whether each graph is correct or incorrect; how many students correctly recognize this; and, if the graph is incorrect, whether students are successful at identifying what needs to be changed to make it correct. • Take time throughout the lesson to ask if any students have questions about the material.

Planned Ending (Closure) • •

20 minutes

Summary Homework

• Briefly summarize and restate the information that was covered during the lesson. • Pass out unit post-test. Students will have 20 minutes to complete it (the use of calculators will not be permitted).

• All students were able to identify that the first warm-up graph was correct. Ten of the 14 students present (Students 4 and 13 were absent) were able to identify that the second graph was incorrect—including Student 12, who initially drew the graph and immediately identified and corrected the error. Although this was based on the cosecant and secant functions rather than tangent and cotangent, the idea behind this was to inform me if any remediation would be necessary; based on the students’ responses, no remediation was needed. • Three of the 14 present students reversed their graphs of tangent and cotangent on the classwork. In one case, this mistake was only made on one graph, suggesting it was a simple error not indicative of a larger problem. However, I observed a pattern of mistakes on the other two students’ assignments. As one of the two students (Student 5) had previously asked how it is determined that the values of tangent and cotangent are

undefined—later admitting she had forgotten the definitions of tangent (ratio of sine to cosine) and cotangent (ratio of cosine to sine)—it appears this was again due to insufficient prerequisite knowledge. At this point in the semester, it would be inappropriate to include any remediation beyond simply reminding the students of these definitions, which is the action that was taken. • To my pleasant surprise, all students were able to correctly identify the asymptotes of the functions (those students who mixed up tangent and cotangent still had the correct asymptotes for their incorrect functions), even on questions involving a phase shift or period change. I had thought this would prove challenging for the students—some students struggled with this when working on cosecant and secant functions, although it appears this may have been due to an incorrect parent graph—and so I was quite pleased to see their mastery of this. Reflection What instructional and/or assessment activities were effective? Why? What instructional and/or assessment activities were less effective? Why? What changes would you make to your instructional and/or assessment procedures that would improve student learning? How did you “think on your feet” to make instructional decisions while you were teaching the lesson? What worked or did not work about these decisions? How did the oral or written feedback you gave students address their needs in relation to meeting the lesson objective? What multicultural considerations emerged during this lesson that you had not anticipated? How would you address those in the future? What insights have you gained from teaching this lesson?

• As was mentioned earlier, one student (Student 5) admitted during the lesson that she had forgotten the definitions of tangent and cotangent. This knowledge is even more basic than the quadrantal values that have given students trouble in this unit, and, as with the values, all students can be reasonably expected to know these definitions at this point in the semester. It would have been wholly inappropriate to include a remediation beyond simply reminding students of the definition, which I did; quite simply, students will not be successful in this or subsequent units if they have not committed these definitions to memory, but this also reflects material that was introduced during the first week of class. • I had hoped to integrate a kinesthetic component during our studies of simply harmonic motion—for example, physically bouncing a ball or demonstrating oscillations of a Slinky—rather than simply taking students through example problems. Unfortunately, due to time constraints I was forced to use the latter method; when I teach this unit in the future, I will make sure to build in increased flexibility so that if remediation becomes necessary, as it did in this unit, I will not have to sacrifice any instructional methods for other components of the unit. I do not feel that taking students through example problems is ineffective, per se; rather, I feel they would have benefitted even more from actual kinesthetic examples.

V. InTASC or COE Standard and Rationale - How does this lesson represent my progress toward gaining mastery of the skills of an effective teacher? This lesson is a good example of InTASC or COE ____ because . . . • InTASC 7 (Planning for Instruction): In all lessons, I based the instruction on the pretest, previous formative assessment, and students’ prior knowledge; however, as this was the final lesson in the unit, this provided me with an opportunity to tie everything together. Despite the exceptionally obscure nature of this lesson (trigonometric functions), I was still able to come up with cross-curricular applications for the students; this also reflects an adjustment of instructional plans, since simple harmonic motion was originally going to be introduced during an earlier lesson but was moved back due to a need for remediation that arose during the initial lesson. This also includes considerations for the absences of two students; although their absences were known in advance, and while I was fortunate enough to have perfect attendance for the previous four lessons, absences are something all teachers must routinely deal with, and so it is helpful to get some early exposure to determining modifications and procedures for catching these students up on the material they have missed. Lastly, this lesson reflects the influence of contextual factors in planning a lesson; although part of the lecture portion of the lesson is devoted to tips for displaying these functions on a graphing calculator, this is done in a manner that is cognizant of the high percentage of FARMS students at the school and thus will not unnecessarily exclude or marginalize students who are less fortunate and do not own graphing calculators. • COE 11 (Use of Technology): All lessons in this unit have made use of the TI Nspire calculators available in the classroom, among its other technological features, but merely integrating technology for technology’s sake is insufficient; it must be properly integrated in a manner that will maximize its benefits for students’ learning. In this lesson, I decided to use overhead transparencies rather than the ActivInspire tablet, with my rationale being that I wanted the ability to display two graphs simultaneously, which can be done on the overhead projector but not with the ActivInspire due to space limitations. In addition, the use of the overhead projector allows me to face the students as I am drawing the graphs to informally assess their reactions—does anyone look particularly lost, or do they all seem to grasp what I’m doing?—whereas the use of the ActivInspire would require me to face the screen so I can see what I am drawing, depriving me of the ability to view students’ reactions during the activity. I also loaded graphs of functions that were going to be demonstrated during the lesson onto students’ calculators before class; while this may seem minor, it ensure that their focus was on transformations of the functions and not simply the functions themselves. Students are not permitted to use calculators on assessments, so preloading the graphs ensured they would use the calculator as a tool to help them visualize what transformations of the function would look like, and not as a crutch to graph a function with no translations. Lastly, as was mentioned earlier, significant care was taken in this lesson to be mindful of FARMS students; I carefully planned the portion of the lesson when I discussed graphing calculator tips to ensure neither students who do not own graphing calculators nor those who do would be singled out.

Section 5: Instructional Decision-Making

Instructional Decision-Making “Wouldn’t it just be the same graph as –cos(x)? Like a reflection across the x-axis?” A few other students nodded in agreement. This comment was in response to my demonstration of how to graph the function cos(–x), which is significantly different from –cos(x). Since the cosine function is an even function, cos(–x) and cos(x) have identical graphs, though they differ from that of –cos(x). I understood the mistake—the students believed they could simply factor out the negative sign as though it were a rational expression—but it reignited a debate with which I had wrestled prior to beginning the unit. In its chapter on trigonometric functions, the students’ textbook includes a description of even and odd functions, but there is no mention of this in the CCPS curriculum guide. After considerable deliberation, I had made the decision to exclude this material from the unit in order to maintain complete alignment with the CCPS curriculum, but my mind kept questioning whether this was the wisest move. Once this misconception by the students came to light, I decided to reverse course and give a brief description of odd and even functions, explaining that –f(x) is not the same as f(–x) for all functions. This will be covered in more detail in a later unit, but I felt it was essential to give students this early exposure to it rather than allowing them to progress further while holding on to this misconception. I was also tempted during this brief tangent to inform students of some techniques with which I was familiar for determining whether a function is even or odd on a TI-83 or TI-84. However, I withheld this information as a nod to the WMHS demographics—namely having the highest percentage of FARMS students in Carroll County. This would certainly have improved learning outcomes for those students who do own their own graphing calculator, but it would also have unfairly singled out the students who are not as fortunate; I felt it was best to entirely withhold the information than to benefit some students at the expense of others.

Section 6: Analysis of Student Learning

ASSESSMENT DATA COLLECTION STUDENT LEARNING Intern:

Ben Cohen

Year: 2013-14

School: Winters Mill High School

Semester: Fall 2013

Grade:

Subject: Trigonometry/Pre-Calculus

STUDENT ID CODE

POSSIBLE POINTS ON PRE ASSESSMENT

POINTS ON PRE ASSESSMENT

PERCENT SCORE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Class Average

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

1 2 0 0 2 0 2 1 2 2 0 2 2 1 0 2 1.19

10% 20% 0% 0% 20% 0% 20% 10% 20% 20% 0% 20% 20% 10% 0% 20% 12%

POSSIBLE POINTS ON POST ASSESSMENT

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

POINTS ON POST ASSESSMENT

3 2 4 6 4 7 2 5 8 3 6 7 4 7 7 6 5.06

PERCENT SCORE

CHANGE PRE TO POST (+ or – Percentage Points)

30% 20% 40% 60% 40% 70% 20% 50% 80% 30% 60% 70% 40% 70% 70% 60% 51%

+20% No change +40% +60% +20% +70% No change +40% +60% +10% +60% +50% +20% +60% +70% +40% +39%

Pretest Item Analysis (Whole Class) Grade: 12 Objectives:

Subject: Trigonometry/Pre-Calculus Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5

Unit: Trigonometric Functions

Intern: Ben Cohen

STUDENT

ALIGNMENT WITH LESSON OBJECTIVES

X X

1 2

3 4

6 7 8 9

X X X

11 12 13 14

X X X

X 8

X 4

15 16 TOTAL CORRECT RESPONSES

Key$%$X$indicates$correct$response$

TOTAL CORRECT RESPONSES

1 2 0 0 2 0 2 1 2 2 0 2 2 1 0 2 19

Post Test Item Analysis (Whole Class) Grade: 12 Objectives:

Subject: Trigonometry/Pre-Calculus Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5

Unit: Trigonometric Functions

Intern: Ben Cohen

STUDENT 1 2 3 5 6

X X X

7 8 9 10 11 12

X X X X X

2 X X X X X X X X X X

13 14 15 16 TOTAL CORRECT RESPONSES

ALIGNMENT WITH LESSON OBJECTIVES

X X X 11

Key$%$X$indicates$correct$response$

X X X 13

X X X

X X

X X X

X X

X X X X 9

X X X X X 11

X X X X X X X X X X X 12

X X

X X X X X

X 11

TOTAL CORRECT RESPONSES

3 2 4 6 4 7 2 5 8 3 6 7 4 7 7 6 81

Discussion of Post Test Item Analysis (Whole Class) •

With the exception of problems 9 and 10 (measuring the second part of the objective from the final class of this unit), which saw no change, all questions on the post-test received more correct answers than their corresponding questions on the pretest, demonstrating that student learning occurred in this unit.

•

Questions related to the lesson 2 objective (horizontal translations) saw the biggest increase; question 2 went from four correct answers to 13, question 3 went from two correct answers to nine, and question 8 went from one correct answer to 12 for an average increase of nine. Given the low achievement on these questions on the pretest, it is clear that students have mastered this objective.

•

Question 4, which pertains to the objective from lesson 1 (vertical translations), also saw a dramatic increase in correct answers, going from two on the pretest to 11 on the posttest. Eleven students also answered question 1 correctly; while eight students correctly answered this on the pretest, three of those students incorrectly answered this question on the post-test, meaning a total of six students demonstrated acquisition of knowledge on this topic and suggesting that these three students’ pretest answers may have been the result of lucky guesses rather than existing content knowledge.

•

Eight students successfully answered question 7, which pertains to the objective from lesson 3 (combinations of translations), representing an increase of seven from the pretest. Notably, the only student who correctly answered this question on the pretest (Student 10) answered it incorrectly on the post-test due to an arithmetic error; as was observed on the pretest, this student sometimes has difficulty with rapid recall and experiences more successful outcomes when he is not under such tight time constraints (although he has not been granted untimed testing); this would appear to be a manifestation of this tendency.

•

Five students correctly answered question 6, which asked students to graph a secant function; while no students correctly answered this question on the pretest, this is still a lower number than would be ideal. A few students made arithmetic errors that led to an incorrect answer; others correctly graphed the parent cosine function but then failed to add in the parabolas for the secant function. While it is clear from the data that some students mastered this objective, additional correct answers would have been ideal.

•

Eleven students correctly answered question 5, which asked students to graph a cotangent function (the first part of the objective from lesson 5). As no students successfully graphed this on the pretest, this demonstrates a strong grasp of the objective. Notably, success on this question was not limited to high-achieving students; five of the 11 students who answered this question correctly scored 50 percent or lower overall.

•

Only one student correctly answered question 9, and none correctly answered question 10; these numbers are unchanged from the pretest. These questions pertained to simple harmonic motion, the second half of the lesson 5 objective. It is clear from this that the instructional time devoted to this topic was insufficient, and it should have been given its own lesson instead of being combined with the tangent and cotangent functions.

DISAGGREGATED ASSESSMENT DATA COLLECTION – FEMALE STUDENTS (Name of Subgroup)

STUDENT LEARNING Intern:

Ben Cohen

Year: 2013-14

School: Winters Mill High School

Semester: Fall 2013

Grade:

Unit: Trigonometric Functions

STUDENT ID CODE

POSSIBLE POINTS ON PRE ASSESSMENT

POINTS ON PRE ASSESSMENT

PERCENT SCORE

POSSIBLE POINTS ON POST ASSESSMENT

POINTS ON POST ASSESSMENT

PERCENT SCORE

CHANGE PRE TO POST (+ or – Percentage Points)

1 2 3 4 5 6 7 8 9 Subgroup Average

10 10 10 10 10 10 10 10 10

1 2 2 0 1 0 2 1 2

10% 20% 20% 0% 10% 0% 20% 10% 20%

10 10 10 10 10 10 10 10 10

3 2 4 7 5 6 4 7 6

30% 20% 40% 70% 50% 60% 40% 70% 60%

+20% No change +20% +70% +40% +60% +20% +60% +40%

1.22

12%

4.89

49%

+37%

Analysis of Disaggregated Data – Female Students: Student Learning •

As with the class as a whole, it is clear that students learned during this unit; the group saw an average increase of 37 percent from pre- to post-test, just below the 39 percent average for the class as a whole. Eight out of nine students (89 percent) saw their scores increase—again comparable to the 88 percent (14 out of 16) for the whole class—with the remaining student (Student 2) experiencing no change in her scores.

•

As with the class as a whole, the average for this subgroup on the post-test (4.89/10) was lower than I had hoped, although the female students’ average was comparable to that of the class as a whole (5.06/10). One of the two students who scored only two out of 10 on the post-test is a member of this subgroup, as is one of the two students who scored three out of 10. Three out of nine students (33 percent)—compared to seven out of 16 (44 percent) for the whole class—experienced increases of 50 percent or more; as the pretest averages were nearly identical (1.19/10 for the whole class, 1.22/10 for this subgroup), this suggests that female students’ learning lagged slightly behind their male peers during this unit. However, it should be noted that two of the four students who scored 70 percent on the post-test, and two of the three who scored 60 percent, are members of this group.

•

The only student in the class who correctly answered one of the simple harmonic motion questions (Student 5) is a member of this subgroup. As a consequence, when considering only students’ performance on the first eight questions, the subgroup’s average actually drops from 49 percent to 48 percent (4.77/10), well below the whole-class adjusted average of 63 percent.

•

One student (Student 13) missed an entire week of class, which included the final day of this unit. Although I had expected this to be reflected in her post-test score—which I expected to be equal to, if not below, her pretest score—in fact she scored four out of 10 on the post-test, a 20 percent improvement over her pretest score and just slightly below the subgroup average. Though certainly not fabulous, it is good to see that any adverse effects from her extended absence—due not only to the material missed on Monday, but also because the material from the unit would not be as fresh in her mind—were limited.

•

In identifying this subgroup, I noted that I expected a wide range of results on the posttest given the heterogeneity of the group with respect to students’ levels of achievement to this point. True to form, of the nine students, four scored in the “upper tier” (six or seven out of 10), three were in the “middle tier” (four or five out of 10), and two were in the “lower tier” (two or three out of 10). This is actually a better result than had been predicted by students’ grades (two grades of “A,” four of “B,” and three of “C”).

•

As I had initially predicted, the female students’ achievement was slightly below that of the class as a whole; thus, male students achieved at a higher level than did females. The mathematics achievement gap between male and female students continues to be an area of tremendous interest for me, and I hope to gather additional data from this and other class in future units to allow me to form a conjecture as to what factors, at least in this class, could be responsible for this and how it can best be rectified.

Pretest Item Analysis Disaggregated Subgroup â&#x20AC;&#x201C; Female Students Grade: 12

Subject: Trigonometry/Pre-Calculus

Objectives:

Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5

Unit: Trigonometric Functions

Intern: Ben Cohen

STUDENT

ALIGNMENT WITH LESSON OBJECTIVES

X X

1 2

4 5

6 7 8 9 TOTAL CORRECT RESPONSES

X X X 5

! Key$%$X$indicates$correct$response$

! !

X X 3

TOTAL CORRECT RESPONSES

1 2 2 0 1 0 2 1 2 11

Post Test Item Analysis Data for Disaggregated Subgroup â&#x20AC;&#x201C; Female Students Grade: 12

Subject: Trigonometry/Pre-Calculus

Objectives:

Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5

Unit: Trigonometric Functions

Intern: Ben Cohen

STUDENT 1 2 4 5 6

X X X X

2 X X X X

9 TOTAL CORRECT RESPONSES

Key$%$X$indicates$correct$response$

! !

X X X

X X 6

X X X X X X

X X

7 8

ALIGNMENT WITH LESSON OBJECTIVES

X X 7

X 4

X X X 6

X X 8

X X X X

X X X

X X X X X X 6

TOTAL CORRECT RESPONSES

3 2 4 7 5 6 4 7 6 44

Discussion of Post Test Item Analysis: Female Students •

With the exception of question 10 (measuring the second part of the objective from the final class of this unit), which saw no change, all questions on the post-test received more correct answers than their corresponding questions on the pretest, showing that student learning occurred in this unit.

•

Question 5, pertaining to the first half of the objective from lesson 5 (graphing a cotangent function), experienced the largest increase; while no students in this subgroup were able to graph this function on the pretest, eight out of nine (89 percent) successfully did so on the post-test. This is a significantly higher percentage than for the class as a whole (11 out of 16, 69 percent), and it is also noteworthy that eight out of the 11 students in the class who correctly answered this question are females.

•

Students in this subgroup were also highly successful at questions assessing the objective from lesson 2 (horizontal translations); seven out of nine students (78 percent) successfully identified a phase shift—compared to three on the pretest—while six out of nine (67 percent) successfully graphed a horizontally transformed cosine function, which no students were able to do on the pretest. Only four out of nine students (44 percent) successfully identified the period of a function, though this was still an increase of three from the pretest. Nevertheless, students in this subgroup were slightly less successful than the class as a whole, which featured success rates of 81 percent (13/16), 75 percent (12/16), and 56 percent (9/16), respectively, on the three questions.

•

Exactly six students were able to identify the vertical translation and amplitude of given functions, questions that measured the objective from lesson 1. Five students successfully identified a vertical translation on the pretest, but two of these students (Students 2 and 8) failed to do so on the post-test, suggesting lucky guesses, and not mastery of the content, were behind their pretest answers. Two students (Students 1 and 2) had identified the amplitude of a function on the pretest, though Student 2 again failed to do so on the posttest. These numbers almost exactly mirror the data for the class as a whole (69 percent, or 11 out of 16, correctly answered these questions on the post-test).

•

Only two students (22 percent) were able to graph a secant function (the lesson 4 objective), though none were able to do so on the pretest. This is significantly lower than the percentage for the class as a whole (5/16, 45 percent); in most cases, arithmetic errors led students to incorrect answers. Similarly, four students were able to graph a sine function given a combination of translations (the objective from lesson 3), compared to none on the pretest; the percentage of students (44 percent) who successfully answered this question is slightly less than the 50 percent of the whole class who did so.

•

The only student (5) in the class to correctly answer question 9 is a member of this subgroup; no students correctly answered question 10, as was the case with the full class. (Neither question received any correct responses from this group on the pretest.) These questions pertained to simple harmonic motion, the second half of the lesson 5 objective. As with the whole class, the results suggest the time devoted to this topic was insufficient.

DISAGGREGATED ASSESSMENT DATA COLLECTION – 80% OR LOWER ON LAST “SPEED TEST” (Name of Subgroup)

STUDENT LEARNING Intern:

Ben Cohen

Year: 2013-14

School: Winters Mill High School

Semester: Fall 2013

Grade:

Unit: Trigonometric Functions

STUDENT ID CODE

POSSIBLE POINTS ON PRE ASSESSMENT

POINTS ON PRE ASSESSMENT

PERCENT SCORE

POSSIBLE POINTS ON POST ASSESSMENT

POINTS ON POST ASSESSMENT

PERCENT SCORE

CHANGE PRE TO POST (+ or – Percentage Points)

1 2 3 4 5 6 7 Subgroup Average

10 10 10 10 10 10 10

1 0 0 2 2 0 2

10% 0% 0% 20% 20% 0% 20%

10 10 10 10 10 10 10

3 4 6 4 3 6 4

30% 40% 60% 40% 30% 60% 40%

+20% +40% +60% +20% +10% +60% +20%

10%

4.28

43%

+33%

Analysis of Disaggregated Data – 80% or Below on “Speed Test”: Student Learning •

Like the full class, it is clear that students in this subgroup learned during this unit; the group saw an average increase of 33 percent from pre- to post-test, slightly below the 39 percent average for the class as a whole. All seven students saw their scores increase from the pretest to the post-test, with increases ranging from 10 percent to 60 percent (a level attained by two students, Students 4 and 11).

•

This subgroup had an average (4.28/10) well below the full class (5.06/10), which is not surprising since these students in particular have struggled with prior material. Students 4 and 11 both scored 6/10 on the post-test, but every other member of the subgroup scored below 50 percent, although the lowest-scoring students in this group (two scored 30 percent) scored higher than the lowest-scoring students in the full class (two scored 20 percent). Only 29 percent of students (two out of seven) experienced increases of 50 percent or higher, compared to 44 percent (seven out of 16) for the full class.

•

It should be noted that this group had a slightly lower average pretest score (10 percent) than the class as a whole (12 percent), and three of the five students who scored 0/10 on the pretest were members of this subgroup. Two of these three students (Students 4 and 11) saw a 60 percent increase from pretest to post-test; the third (Student 5) increased by 40 percent. No other students improved by more than 20 percent, suggesting that their lack of knowledge of trigonometric values continues to hold them back.

•

When adjusting students’ scores to count only the first eight questions, the average score increases to 54 percent, well below the full-class adjusted average of 63 percent. Again, this is not surprising considering the difficulties these students have previously encountered in the class.

•

On the surface, Student 11’s success in this unit was not entirely unexpected, since she had the highest score in this subgroup on the last “speed test” (37/48, 77 percent). However, Student 4 had the fourth-lowest score in the class (22/48, 46 percent) but tied her for the highest score on the post-test; on the other hand, Students 1 and 13 both had the second-highest “speed test” score in the subgroup (36/48, 75 percent) but scored just 30 percent and 40 percent, respectively, on the post-test. These scores are what would be expected from students who have not demonstrated adequate command of trigonometric values—they would be able to identify the period, phase shift, vertical translation, and amplitude of a given function but not graph it. It appears, then, that Students 11 and 4 benefitted significantly from the remediation in this unit, while the remaining five students are still struggling with trigonometric values.

•

Above all, the data show the importance of mastering the early material due to the cumulative nature of mathematics. Students have been warned throughout the semester that they will struggle in this class if they do not know their trigonometric values, and this unit serves as another reminder of that. Two students appear to have benefitted from remediation built into the unit, and I will continue to build remediation into future units in hopes of benefitting additional students.

Pretest Item Analysis Disaggregated Subgroup – 80% or Less on Last “Speed Test” Grade: 12

Subject: Trigonometry/Pre-Calculus

Objectives:

Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5

Unit: Trigonometric Functions

Intern: Ben Cohen

STUDENT

ALIGNMENT WITH LESSON OBJECTIVES

1 2 3

6 7 TOTAL CORRECT RESPONSES

X 1

! Key$%$X$indicates$correct$response$

! !

X 2

TOTAL CORRECT RESPONSES

1 0 0 2 2 0 2 7

Post Test Item Analysis Data for Disaggregated Subgroup – 80% or Less on Last “Speed Test” Grade: 12

Subject: Trigonometry/Pre-Calculus

Objectives:

Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5

Unit: Trigonometric Functions

Intern: Ben Cohen

STUDENT 1 2 4 5 6

X X X X

2 X X X X

! Key$%$X$indicates$correct$response$

! !

X X X

X X

7 TOTAL CORRECT RESPONSES

ALIGNMENT WITH LESSON OBJECTIVES

X 3

X 4

X X 3

X X X 4

TOTAL CORRECT RESPONSES

3 4 6 4 3 6 4 30

Discussion of Post Test Item Analysis: Below 80% on “Speed Test” •

With the exception of question 10 (measuring the second part of the objective from the final class of this unit), which saw no change, all questions on the post-test received more correct answers than their corresponding questions on the pretest, demonstrating that student learning occurred in this unit.

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Students were most successful at question 2 (identifying a phase shift), which pertains to the lesson 2 objective; five out of seven students in this subgroup (71 percent) correctly answered this question, compared to two on the pretest. Two other questions pertained to this objective; four out of seven (57 percent) correctly graphed a horizontally translated cosine function, which only one student (Student 10) was able to do on the pretest, while only three students (43 percent) successfully identified the period of a function, compared to one on the pretest. These percentages are lower than those of the class as a whole (81 percent, 75 percent, and 56 percent, respectively), which is not surprising since these students have struggled relative to their peers throughout the semester.

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Exactly four students correctly answered the two questions pertaining to the lesson 1 objective (vertical translations); notably, only one of these four students (4) correctly answered both questions. One student correctly answered each question on the pretest (Students 13 and 5, respectively), so the increase is significant. Again, the percentage of students in this subgroup is lower than that for the full class (69 percent—11 out of 16— for each). Notably, Student 13 correctly identified a vertical translation on the pretest but not on the post-test, suggesting she may simply have guessed correctly on the pretest; Student 5 correctly identified the amplitude of a function on both the pre- and post-test.

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Three students were able to graph a sine function given a combination of translations, which only one student (Student 10) successfully completed on the pretest; this pertained to the objective from lesson 3. This percentage (43 percent) closely parallels the 50 percent of students in the full class who correctly graphed this function.

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Question 5, which measured the first half of the lesson 5 objective—graphing a cotangent function—saw a significant increase: Four students in the subgroup were able to complete this on the post-test, compared to none on the pretest. However, the 57 percent of students answering this question correctly still pales in comparison to the 69 percent doing so in the full class. Students were less successful at the lesson 4 objective (graphing a secant function); this question received the third-fewest correct answers (two) of any question on the post-test, though this was still an improvement over the pretest, on which no student responded correctly. The 29 percent of students in the subgroup closely parallels the 31 percent of students in the whole class who answered correctly.

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The only student in the class who correctly answered question 9 (Student 5) is part of this subgroup; no students correctly answered question 10, as happened with the whole class. (Neither question received any correct responses from this group on the pretest.) These questions pertained to simple harmonic motion, the second half of the lesson 5 objective. As with the whole class, the results suggest the time devoted to this topic was insufficient.

Section 7: Evaluation and Reflection

Reflection and Self-Evaluation In this unit, the objective at which students were the most successful was the lesson two objective: “Students will be able to graph horizontal translations of the sine and cosine functions and will be able to identify the period and phase shift of these functions.” Thirteen out of 16 students successfully identified the phase shift of a function on the post-test; 12 more were able to graph a horizontally translated cosine function. In total, 75 percent of the responses to the three questions pertaining to this objective were correct, the highest percentage for any of the five objectives from the unit. There are numerous possible reasons for this success. First and foremost, I must give credit to the “ ‘X’ marks the spot” method of instruction that my mentor suggested I consider for this lesson; he had used it previously for over 20 years, with great success. Although this differed from the instructional method I had planned to use, it proved to be enormously successful, as evidenced by students’ achievement not only on the post-test, but on the formative assessments as well. The success of this teaching method showed the importance of being willing to accept suggestions as a teacher, and I will be sure to integrate it into future units. In addition, I made sure to use the proper vocabulary (“period” and “phase shift”) more frequently during this lesson than during the previous one, when the terms “vertical shift” and “amplitude” were not introduced until late in the lesson. Students were less successful at identifying both a vertical shift and amplitude than at a phase shift but were generally able to graph changes to these values; as such, it appears their relative lack of success at identification was due to insufficient familiarity with the vocabulary. Again, this is something that I will need to take into consideration in future units. It is also possible the activities I had planned at the start of the unit helped with students’ understanding of these concepts. During the first lesson, I displayed the sound waves of three

guitar solos to help students understand the concepts of amplitude and period; the following day, I used sound waves of two items familiar to all students—the WMHS class bell and fire alarm— to reinforce these concepts. I made it a point throughout the lesson to ensure students were not acquiring their knowledge in a vacuum, and while cross-curricular connections were included in all lessons, they were the focal point of these first two classes. It is tempting to declare here that they should have been the focal point of the remaining three classes as well, but due to the nature of the material—cartography problems involving the cosecant and secant functions, for example, require a level of mathematical knowledge far beyond what can be reasonably expected from high-school students—this would not have been feasible. Lastly, while many students struggled in this unit due to a lack of knowledge of trigonometric values, this knowledge is not necessary to determine period and phase shift, meaning these students were not put at a disadvantage. In addition, on the graphing question pertaining to this objective, it is possible some students may have gotten it right by graphing it incorrectly. This may seem wholly counterintuitive, but a horizontally translated cosine graph, in many instances, resembles a graph of the sine function; given that many of these students tend to mix up their values of sine and cosine, it is possible this may have led them to the correct answer. While this obviously does not indicate mastery of the content, it must be considered as well. The ESL provided me with considerable valuable experience in the areas of lesson planning, delivery of instruction, and class management. Insights gleaned from this unit include: Organization and preparation are crucial. It is now clear to me why the MAT program requires that all lesson plans and materials be submitted well in advance of teaching the lesson— planning the unit as you go or attempting to “wing it” will almost inevitably lead to failure. Particularly in high school, students will realize quickly when a teacher is unprepared—I did this on occasion as a student—and in many cases, this will lead to a concomitant drop in their effort.

Ensuring in advance that all instructional materials are ready for a lesson will ensure optimal learning outcomes for students. Teachers must be able to adjust “on the fly.” The previous insight does not mean that a plan should be determined for every unit at the beginning of the school year and then rigidly adhered to with no room for alterations. In addition to interruptions such as snow days, students may grasp some material more quickly than expected, or may need more practice than had originally been planned. In the case of this unit, it became necessary to devote part of one lesson to remediation of prerequisite knowledge; I also observed this more recently, when the planned topic (inverse trigonometric functions) was pushed back one day to allow students an additional day of practice with trigonometric identities. My mentor has warned me that this must be kept at a reasonable level—for example, the same topic should not be covered in three consecutive lessons—but it must be kept in mind to ensure students’ needs are met. Don’t assume all students possess prerequisite knowledge. During this unit, students were expected to have fully committed to memory values of the six trigonometric functions for specified angles. However, it became clear during the lesson—as was also indicated by the last “speed test,” which was designed to measure students’ mastery of this knowledge—that a significant portion of the class had gaps in its knowledge of this topic. Based on an analysis of “speed test” grades, I had prepared for this and had built remediation into the unit, but I had not expected students to encounter difficulty adding fractions, or need to be reminded of the definition of tangent—problems that arose later in the unit. Although these were items that required nothing more than a simple reminder, I need to keep in mind in future units not to take for granted that students will have a thorough understanding of the prerequisite content knowledge; again, while it is impossible to plan for every contingency, a situation may arise

where a remediation more substantial than a simple reminder is necessary, and I will need to be adequately prepared for this to ensure optimal outcomes for my students. Be open to accepting suggestions. Prior to the lesson on horizontal translations, my mentor shared with me a method he had used previously, with great success, which he named “ ‘X’ marks the spot.” Although this differed from the way I had planned to introduce this material, it proved to be enormously successful; student mastery of this day’s objective, as measured by the post-test, was unmatched by any other lesson. I have always felt that teachers are constantly learning—both from their students and from each other—and that the day I start to feel that there’s only one way to teach a topic is the day I cease being an effective educator. Teaching is a process of constant evaluation, and even if I’ve been teaching something the same way for 20 years, it doesn’t mean that I shouldn’t be open to suggestions on a different method. Technology is your friend—if used wisely. The amount of instructional technology available to educators today is incredible—from SMART Boards to TI Nspire calculators—but must also be used in a manner that will ensure optimal learning outcomes for students. Merely integrating technology for technology’s sake will likely not actually improve student learning. For example, during two lessons, I intentionally went to a “low-tech” alternative (overhead transparencies) rather than the more sophisticated ActivInspire tablet since I felt the former method would lead to better outcomes for students (for reasons that have previously been described in detail). Include applications throughout the unit. The most common refrain I have heard from students throughout this year is, “When are we ever going to use this?” In addition to including a description of numerous applications of trigonometric functions in the first lesson of this unit, I made sure to include an application in each subsequent lesson; while some (maritime navigation; cartography) required a level of mathematics knowledge far beyond the scope of this class and

thus had to be limited to descriptions, others (such as simple harmonic motion) played central roles in this unit. In addition to ensuring continuing student interest and engagement, this ensures that students are not learning these mathematical concepts in a vacuum but are instead aware of the roles they play in their everyday lives. Take time to get to know students. While class is of course not social hour, and it is crucial to maintain the distinction that you are the students’ instructor, not their friend, I have realized it is similarly important to ensure the students do not see you as merely a soulless robot who teaches them and is incapable of doing anything else. Many of my college math professors suggested that I view their classes as a “two-way conversation,” and I have tried to bring the same concept to my classes; my goal is to work with students to help them succeed, rather than simply talking to them about the material. I feel it becomes easier to accomplish this if I have already built up a rapport with the students—though again, making sure they do not start to view me as their contemporary—and I have made this a priority throughout the academic year. One-on-one tutoring can be highly beneficial for students. This does not have to be simply the instructor working with students during or after class; it can also include peer tutoring and small-group work. I was fortunate that my classroom included two instructors during this unit, but I will not always have this luxury; as such, I must be sure to include opportunities for one-on-one work (including student-to-student activities) throughout the lessons in future units. This last insight segues into one professional development goal that emerged from the ESL, that being one-on-one tutoring. I am currently working on a research project measuring the effectiveness of one-on-one tutoring on students’ learning outcomes; the literature suggests it can have a significant positive effect. Although I worked as a tutor in college and have been using FLEX time to work individually with students as necessary, I am always looking to improve my skills in this area; to that end, I am volunteering as a tutor in the Falcon Learning Lab after

school on internship days and plan to do so for the duration of my time at WMHS. In addition, as I mentioned earlier, peer tutoring and small-group work can also be highly beneficial for students; over the course of this year, I hope to build up my knowledge base regarding the best practices for implementing these strategies in the mathematics classroom, both through a review of the literature and by taking advantage of available professional development opportunities such as the MCTM and NCTM conferences, the latter of which I was able to attend this year owing to it being held in the Baltimore Convention Center. Another area in which I hold lofty goals pertaining to professional development is the integration of technology. As I detailed above, instructional technology can be an invaluable resource—if it is deployed properly. I am fortunate enough to teach in a classroom that has a supply of TI Nspire calculators; although I made sure to integrate these into the lessons, they were not a focal point. As a means of expanding my knowledge base as to how to fully harness the power of these calculators, I attended a TI workshop a few weeks ago; while this was, unfortunately for me, held on the Saturday following ITE week, it gave me numerous insights into how to best integrate these calculators in my classroom, and I am planning to make them a focal point of the units I will teach during the second ITE this semester. (I have also been asked to share the knowledge and insights I acquired at this workshop at an upcoming WMHS math department meeting; I will also use this time to survey the faculty and see if they have any ideas on best practices for these calculators.) I also attended multiple sessions focused on technology at this year’s NCTM conference, including a workshop led by the creator of Geometer’s Sketchpad that focused on its integration throughout the math curriculum; I will continue to attend local workshops and conferences and also participate in national and international webinars—and combine this with a rigorous review of the literature, including Mathematics Teacher magazine—to continue learning about best practices for technology integration.