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Evidence of Student Learning Benjamin Cohen Towson University EDUC 798 Internship II with Seminar Trigonometry/Pre-Calculus Grades 11-12 Winters Mill High School March 2014


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

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

Learning Objectives Learning Objectives .......................................................................................................................36

Assessment Plan Assessment Alignment Summary ..................................................................................................41 Pretest.............................................................................................................................................42 Post-Test ........................................................................................................................................44 CCPS Summative Assessment.......................................................................................................46 Formative Assessments..................................................................................................................48 Identification of Subgroups............................................................................................................56

Lesson Plans Pretest Item Analysis (Whole Class) .............................................................................................58 Pretest Item Analysis (Female Students) .......................................................................................60 Pretest Item Analysis (Students Below 90% Attendance) .............................................................62 Lesson Plan 1 .................................................................................................................................64 Lesson Plan 2 .................................................................................................................................72 Lesson Plan 3 .................................................................................................................................80 Lesson Plan 4 .................................................................................................................................88 Lesson Plan 5 .................................................................................................................................96


Classroom Management Matrix ...................................................................................................103

Instructional Decision-Making Instructional Decision-Making ....................................................................................................107

Analysis of Student Learning Assessment Data Collection (Whole Class) ................................................................................110 Pre-/Post-Test Item Analysis (Whole Class) ...............................................................................112 Disaggregated Assessment Data Collection (Female Students) ..................................................116 Disaggregated Assessment Data Collection (Students Below 90% Attendance) ........................118 Pre-/Post-Test Item Analysis (Female Students) .........................................................................121 Pre-/Post-Test Item Analysis (Students Below 90% Attendance) ...............................................125

Evaluation and Reflection Evaluation and Reflection ............................................................................................................130


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 this unit on solving oblique triangles. 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 trigonometric 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 58-63 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 fourth lesson plan from this unit. This lesson was the first in a two-part series introducing students to the applications and historical context of oblique triangles, so I felt it was particularly important to keep in mind the high percentage of FARMS students at Winters Mill when planning it. All readings used in this lesson were carefully selected to ensure they represented topics that all students, regardless of socioeconomic status, would be able to relate to, such as the model of the solar system and the development of calculators that they would be given the opportunity to use later in class. The goal was to ensure all students would fully develop an appreciation for the power of oblique triangles and achieve to the best of their abilities, rather than getting bogged down in unfamiliar concepts that would have only served to make them feel disconnected from my instruction. In addition, the discussion portion of this lesson was twofold; not only was it designed to increase students’ knowledge and


comprehension of the concepts discussed in this unit, but I was also eager to see students’ diverse contributions to the class and gain an understanding of how their unique backgrounds and life experiences shaped their approach to this activity and the relationships and connections they were able to draw. 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. In addition, student engagement is crucial to the success of a lesson, particularly in a field such as math, which, as I always like to say, is learned by doing, not by seeing. My artifact for this standard is the lesson plan from my fifth ESL lesson, specifically the measuring activities that took place throughout the school building. This was a highly engaging activity that ensured students were active participants, rather than spectators, in the learning process. While I make student engagement a focal point of every lesson, this allowed students to move beyond simple calculations and answers to questions and truly embody the “doing� part of my philosophy on math education. As indicated in the lesson plan, I also instructed students to think about additional applications of such an activity on a larger scale, such as measuring the area of a triangle with its vertices at three regional cities, to keep them actively engaged and maintain a constant thought process. In addition, students were placed into groups of three to ensure they had the opportunity to work cooperatively to achieve the goals of


this lesson. The emphasis on positive social interaction and working together for a common goal make this an ideal representation of standard three. All groups were able to successfully determine the area of each triangle (within a reasonable margin of error); in addition, a number of students remarked that this was a very fun and engaging activity—particularly salient feedback since this was the last day of the first full five-day week of the semester (all prior weeks had been shortened or interrupted due to weather). This artifact helps represent the “planning instruction” and “teaching” components of the CPTAAR cycle. A well-planned class includes not only an active, engaging instructional component, but also a series of 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. This lesson certainly meets that goal; rare is the lesson when students can venture outside the classroom for an activity that truly is “hands-on.” Certainly, it should be noted that this would not be possible in all classes—I could not supervise over 30 students in the hallway at once, and a foundational class would be more significantly prone to misbehavior or other complications—but when the opportunity to integrate an activity such as this one presents itself, an effective teacher will certainly take advantage of it. The lesson plan is included on pages 96-102 of the ESL. The descriptions of the measuring activities are electronically highlighted and can be found on pages 99-100.


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 have chosen the article and discussion on the Bermuda Triangle from the final lesson of this unit. The article continues the theme of drawing cross-curricular connections to the material being studied in the unit. 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?” I asked students to read the article and think about how some of the key questions relate to what they had previously studied, such as why so many planes and ships disappear in the Bermuda Triangle without a trace despite our advanced searching capabilities (using Heron’s


formula, we can find that the area is over one million square miles—a really, really big area to search). I also wanted to use problems to which students can legitimately relate in their everyday lives, rather than, say, cell tower triangulation. This also breaks up the pattern of problemsolving by asking students to engage in critical thinking and synthesize information, providing them with an additional means of demonstrating their content knowledge and improving their learning outcomes by allowing them to view and think about the content in a new light. 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)—when introducing this particular topic, for example, I made no mention of students potentially taking a cruise in the Caribbean to avoid excluding students who could not relate. The lesson plan can be found on pages 96-102 of the ESL; the discussion description is electronically highlighted and located on page 99.


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. For this standard, I chose as my artifact the lesson plan from the fourth day of the unit, which includes a directed reading lesson featuring passages from a history of mathematics textbook. This artifact fits standard five because it focuses on content-area reading strategies to support the development of students’ literacy in mathematics, an item that this standard specifically addresses. In addition, the reading activities focused on having students draw connections between the material in this unit and their past classes, such as how the Laws of Sines and Cosines explain why a triangle must have exactly 180 degrees. The lesson also focused on additional applications of these techniques to real-world problems, such as the development of our current model of the solar system and the development of new technology.


The content-literacy activities had a positive impact on student learning, though not as positive as I had originally hoped. The readings were on too high a level for students to understand without supports beyond what I had provided; as a consequence, only one student was able to fully complete the graphic organizer. With that said, all students’ work revealed deep insights and high-level thinking about the topics at hand, showing that this was a beneficial activity—they simply did not have enough time to put everything together. This artifact aligns with the “planning instruction” and “teaching” components of the CPTAAR cycle. A frequent criticism of mathematics instruction is the relative dearth of reading and writing activities, so content-literacy strategies, when implemented appropriately, can be of enormous benefit to students. However, it is necessary with these activities to be prepared to adjust “on the fly” and express the topic in a different form if necessary to improve students’ comprehension. This is particularly true of content-literacy activities in math: the readings will typically be at such a high level that it is almost inevitable some adjustments will be necessary. The lesson plan can be found on pages 88-94 of the ESL.


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 first day of this unit. In addition to traditional formative assessment techniques such as classwork and homework assignments, I integrated numerous informal assessment techniques. The Geometer’s Sketchpad demonstration, in addition to providing an interactive means to show students the derivation of the Law of Sines, served as an assessment technique by allowing me to observe the extent to which students had retained relevant content knowledge from Geometry. I also complemented this demonstration with a series of algebraic derivations similar to those required to solve problems involving the Law of Sines; in addition to using students’ familiarity and comfort level with these manipulations in my planning for subsequent lessons, I was ready to make any necessary instructional modifications—such as including remediation—during this lesson based on my observations of students. It is also key to note that these various assessment techniques


provided me with immediate feedback during the lesson—while the pretest administered at the start of the lesson also provided a significant amount of data, this data was not available until after the lesson, once I had gotten a chance to grade and analyze the pretests. This instant feedback made it possible for me to adjust instruction as necessary to meet the needs of students. 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 64-70 of the ESL; the relevant descriptions of the assessment techniques are electronically highlighted and can be found on pages 64 and 66. A printed copy of the Geometer’s Sketchpad demonstration is available on page 71.


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. 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. In addition, the best teachers set rigorous learning goals for their students, encouraging them to constantly think deeper and push themselves to extend their knowledge and understanding. My artifact for this standard is the lesson plan from the second day of this unit. As I explained to students throughout the unit, my feeling is that this was one of the three most difficult units they will study this year, and the most difficult topic in this unit is the derivation of the Law of Cosines—the topic of the second day’s lesson. I intentionally integrated an extremely challenging derivation as a means of pushing the students to extend their knowledge and think deeply about the topic; unfortunately for me, this ended up being too difficult for the students, and I adjusted my plans for the second-period class to make this information more accessible. However, this lesson did not focus solely on a rigorous derivation—I also included a portion dedicated to introducing students to the real-world applications of the Laws of Sines and Cosines, including asking students to think of an application. I was eager to see how students’ diverse


backgrounds and cultures shaped their thoughts on this topic; I also took care when planning applications to introduce that I only included items to which students could unquestionably relate. (A discussion on a cruise ship evading icebergs, for example, would likely have excluded lowSES students who had never been on a cruise ship.) The combination of rigor and diversity as planning considerations makes this an ideal illustration of standard seven. This artifact reflects the “understanding the context for learning” and “planning instruction” components of the CPTAAR cycle. As was stated earlier, 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 throw-in to the lesson rather than a seamlessly integrated component of it. It is also crucial to make sure each lesson represents an appropriate level of rigor; students will not benefit if they are not being challenged, nor will they learn if all of the material is being taught at a level well above their heads. This artifact can be found on pages 72-78 of the ESL. Details on the considerations described above are electronically highlighted and located on pages 73-76.


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 making content accessible and meaningful for students, rather than simply being taught “because it’s part of the curriculum.” My chosen artifact for this standard is the lesson plan from the second day of this unit. Although this lesson necessarily included a heavier lecture component than the subsequent classes, since it served as one of two foundational lessons at the start of the unit, I attempted to employ a number of different instructional strategies to ensure students remained engaged throughout the lesson. When planning the lesson, I included elements designed to appeal to all types of learners; the lesson featured technology (a demonstration in Geometer’s Sketchpad), algebraic derivations (deriving the formula to find an angle measure), connections to past material (the Pythagorean theorem is a special case of the Law of Cosines), ample opportunities to practice the skills learned in this lesson, and even an interactive demonstration and sports analogy at the beginning (the distance a pitcher must throw the ball to first base when fielding a bunt). I also focused on making the content meaningful for students: as is my custom, part of the


lesson was devoted to an introduction to, and discussion of, real-world applications of the techniques introduced in this lesson. All examples were chosen to appeal to this specific group of students—such as passes of a soccer ball and open-field football tackles for the athletes and the distance between military bases for the JROTC cadets. I also asked all students to think of an example in which they could apply these trigonometric laws to their everyday lives; in theory, at least, these examples should be quite meaningful for the students providing them. 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 addition, the emphasis on making content meaningful dovetails with the “context for learning” portion of the cycle—it is crucial to have a solid grasp of students’ backgrounds and cultures when planning instruction that will allow them to truly relate to the topic at hand. This artifact can be found on pages 72-78 of the ESL. Details of the example discussion are electronically highlighted and can be found on pages 74 and 76.


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: my mentor’s feedback from the second lesson of this unit, and the following day’s lesson plan, showing adjustments made in response to this feedback. The decision to omit a Geometer’s Sketchpad demonstration from the third lesson was not one that I took lightly, given the considerable extent to which I typically integrate Sketchpad demonstrations into my lessons as well as its demonstrated utility from the first lesson plan. After consulting with my mentor, however, I came to the conclusion that the derivations related to the Law of Cosines during this lesson were too abstract for students, and including a similar demonstration in the next lesson would have hindered rather than helped the students. Dismissing this suggestion as “unimportant” or forging on for the sole purpose of


integrating technology for technology’s sake 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 again make clear that I constantly strive to reflect on and improve my teaching strategies and am willing to consider and integrate any piece of advice, no matter how consequential it may be for my existing plans. I replaced the Sketchpad illustration with a simpler algebra-based derivation, which was far more accessible for students. 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, a practice I began after my first ITE during the fall semester and now do following each lesson I teach—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 day’s lesson. The lesson plan from day 2 can be found on pages 72-78; the reflection on the Sketchpad demonstration (including my mentor’s feedback) is electronically highlighted and is located on page 77. The following day’s lesson plan is included on pages 80-86; the impact of the previous day’s reflection on my planning is also electronically highlighted and can be seen on page 81.


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 teachers’ 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 one’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 the lesson plan from the final day of this unit. This includes a pair of activities that I developed after collaborating with my mentor and other members of the Winters Mill math department. First, my mentor and the other trigonometry teachers in the department gave me the idea for the measuring activities as a strong means of involving all learners and making the content accessible for everyone; this was also a very helpful activity for allowing students to become actively involved in their own learning. In addition, my mentor and I decided “on the fly” to include a brief (roughly 10 minutes) discussion of the metric system when students returned to the classroom after measuring the triangles. He had observed a number of students questioning why there were two sets of numbers on the measuring tape and brought this to my attention; after a brief consultation, we decided to


introduce students to the metric system and also discuss why it might be preferable to the imperial system of measurement, and why the U.S. is the only developed country that has yet to switch to the metric system. Both of these activities were developed through collaboration with other teachers, making them ideal illustrations of standard 10. This artifact mirrors the “planning” and “teaching” components of the CPTAAR cycle. Collaboration with other school professionals during the planning process could include classroom management strategies, best practices for teaching that day’s topic (including materials that had been successful previously), technological or multimedia resources, or strategies for working with diverse learners; ultimately, meaningful collaboration will lead to greatly enhanced learning outcomes for students. In addition, successful implementation of these strategies is crucial to the success of the lesson; all the planning in the world can go to waste if it is not effectively put into practice. This artifact can be found on pages 96-102 of the ESL. Information regarding collaboration as it pertains to these activities is electronically highlighted and can be found on page 102.


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 Geometer’s Sketchpad demonstration from the first lesson of this unit. I always attempt to include a significant technological component in each lesson, and this one was no exception; in addition, Geometer’s Sketchpad is a particular favorite program of mine given its utility across the mathematics curriculum. As an added bonus, this demonstration, in addition to being interactive and allowing students to clearly visualize the derivations of the Law of Sines, served as an informal assessment technique allowing me to measure the extent to which students had retained knowledge from Geometry and in turn how much remediation (if any) would be necessary. I was able to do this by observing the frequency and accuracy of students volunteering to answer questions pertaining to the demonstration.


This artifact closely parallels the “planning instruction,” “teaching,” and “assessment” 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 is possible to use such a demonstration (or other technology) as a non-traditional, informal assessment technique. Rather than a pretest or other similarly structured assessment that must be done separately, assessment techniques such as this can be integrated directly into the lesson and thus maximize instructional time. This artifact can be found on page 71 of the ESL. The corresponding lesson plan can be found on pages 64-70 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 Maryland’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-$12,230—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 WMHS 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 LCD 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 section of Trigonometry/Pre-Calculus that I will be teaching takes place daily during mod 1, which meets from 7:30-8:50 a.m. There are 16 students enrolled in this section of 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 are 10 students in the class with “medical alerts,” but according to the instructor, these pertain to conditions that do not affect their learning outcomes, other than the possibility of extended absences. There are only three minority students in the class: two African-Americans (one male, one female) and one Hispanic male. Eleven members of the class are 12th graders who have previously taken Intermediate Algebra, Geometry, and Algebra II, in that order; the remaining five are 11th graders with the same course sequence. Seven students received grades of “A” in Algebra II, six received a “B,” and three 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 TI-Nspires presents significant opportunities for the integration of technology. I have also made extensive use thus far of demonstrations in Geometer’s Sketchpad on my personal computer and plan to continue doing so throughout this unit. 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 finding the linear and angular speeds of Sally Ride and John Glenn’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 three minority students 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 three students 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 modern-day Baltimore County) into my lessons. In fact, the first half of one lesson in this unit will be devoted specifically to an exploration of these contributions. As was mentioned earlier, all students in this class have previously taken Intermediate Algebra, Geometry, and Algebra II, in that order, according to the CCPS curriculum guide. Students will be expected to be familiar with sine, cosine, and tangent values for quadrantal and special acute angles, which were covered in an earlier unit; they will also be expected to recall the definitions of sine (opposite/hypotenuse), cosine (adjacent/hypotenuse), and tangent


(opposite/adjacent) for a specified angle. It is also expected that they will recall the Pythagorean theorem (a2 + b2 = c2) from Geometry, although this is not crucial to success in this unit, which is primarily an expansion of their knowledge of right-triangle trigonometry. Whereas they previously were only familiar with techniques to solve a right triangle (such as the Pythagorean theorem), they will now be introduced to the methods of applying these same techniques to solve an oblique triangle (one without a right angle). It is also expected that students will recall the relationship between angle measure and quadrant; although a minor detail, this is enormously significant when using these solving techniques. An analysis of the pretest suggests that students have retained a significant amount of content knowledge from Geometry and the previous unit on solving right triangles but, as expected, are unfamiliar with how to apply these methods to solve oblique triangles. Fifty percent of students (eight out of 16) correctly solved multiple-choice problems involving the Law of Sines and the SAS case of the Law of Cosines (though none correctly answered a freeresponse question on the same skill set); both of these are extensions of students’ existing knowledge, and these results do seem to indicate there is a solid foundation of knowledge to build off. At the same time, no students were able to solve any problems involving three given sides (SSS); while students are aware from their previous studies that SSS is a means of verifying triangle congruence, and that the length of the third side in a right triangle can be found by using the Pythagorean theorem, they have had no prior experience translating this knowledge to oblique triangles. There were quite a few students (six out of 16; 38 percent) who successfully found the area of an oblique triangle using the sine formula, though it appears based on their comments that this was a byproduct of successful guessing. Overall, it is clear that significant time will need to be devoted to these topics and their derivations. 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: 11/12 Class: Trigonometry/Pre-Calculus Unit: Solving Oblique Triangles Triangles are one of the oldest and most frequently studied objects in all of mathematics, dating back to Ancient Greece. Oblique triangles are but a small part of this enormous field of study, but they also constitute one of the most significant ones. In this unit—which is arguably one of the three most difficult they will study in this course—students will not only learn the mathematics behind solving oblique triangles, but also gain a greater appreciation for their role throughout history and the myriad ways in which they have affected our everyday lives, such as through the development of the heliocentric model of the solar system in the 16th century. While the primary focus will of course be on mathematics, this unit also marks the beginning of an effort for the rest of my time at WMHS to use this class to help students tie together everything they have studied in their 3-4 years of mathematics; there are few better ways to do so than by studying a topic that forms the backbone of countless ancient and contemporary advances in mathematics. Lesson 1 Topic: Law of Sines Curriculum Objective: Given AAS, ASA, or SSA, use the Law of Sines to solve oblique triangles. Maryland Content Standards: 2.2.2.1 The student will apply the Law of Sines and the Law of Cosines to solve problems involving oblique triangles. Student Objective: Given two angles and a side (included or non-included) or two sides and the included angle of an oblique triangle, students will use the Law of Sines to solve the triangle (i.e., find all remaining side and angle values). Lesson 2 Topic: Law of Cosines (SAS) Curriculum Objective: Given SAS, use the Law of Cosines to solve oblique triangles. Maryland Content Standards: 2.2.2.1 The student will apply the Law of Sines and the Law of Cosines to solve problems involving oblique triangles. Student Objective: Given two sides of an oblique triangle and the included angle, students will use the Law of Cosines to solve the triangle.


Lesson 3 Topic: Ambiguous Case of the Law of Cosines Curriculum Objective: Given SSS, use the Law of Cosines to solve oblique triangles. Maryland Content Standards: 2.2.2.1 The student will apply the Law of Sines and the Law of Cosines to solve problems involving oblique triangles. Student Objective: Given the three side lengths (but no angle measures) of an oblique triangle, students will use the Law of Cosines to solve the triangle. Lesson 4 Topic: Area of Oblique Triangles (Heron’s formula) Curriculum Objective: Given SSS, find the area of an oblique triangle using Heron’s Formula. Maryland Content Standards: 2.2.2.1 The student will apply the Law of Sines and the Law of Cosines to solve problems involving oblique triangles. Student Objective: Given the three side lengths of an oblique triangle, students will use Heron’s formula to find the area of the triangle. Lesson 5 Topic: Applications of Oblique Triangles Curriculum Objectives: Given SAS, ASA, or AAS, find the area of an oblique triangle. Maryland Content Standards: 2.2.2.1 The student will apply the Law of Sines and the Law of Cosines to solve problems involving oblique triangles. Student Objective: Given two sides and one angle or two angles and one side of an oblique triangle, students will find the area of the triangle. They will also solve problems involving applications of areas of oblique triangles.


Section 3: Assessment Plan


Assessment Alignment Summary The following chart provides a summary of the assessment plan for the Solving Oblique Triangles 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 objective—for example, students were introduced to solving triangles given two sides and the included angle in lesson two; this was also assessed in lesson five, which asked students to find the area of an oblique triangle using the sine formula. Note: There is only one Maryland Content Standard that applies to this unit, as stated in the CCPS curriculum guide; for brevity, it is listed only by its number in the following chart. The full standard is as follows: 2.2.2.1 The student will apply the Law of Sines and the Law of Cosines to solve problems involving oblique triangles.


Trigonometry/Pre-Calculus Solving Oblique Triangles – Pre-Test

Name: ______________________

Part I. Find the missing parts of the triangle. 1. C = 122.6o, a = 8.00 m, b = 8.16 m A) c = 15.3 m, A = 45o, B = 81o

B) c = 17.1 m, A = 30.3o, B = 27.1o

C) c = 14.2 m, A = 28.3o, B = 29.1o

D) c = 20 m, A = 26.3o, B = 31.1o

2. A = 30o, a = 3, c = 6 A) B = 60o, C = 90o, b = 3

B) B = 60o, C = 90o, b = 3 3

C) B = 90o, C = 60o, b = 3 3

D) B = 90o, C = 60o, b = 3

3. A = 65.3o, a = 2.15 km, b = 2.25 km

4. a = 163 yds., b = 182 yds., c = 328 yds.

Part II. Find the area of triangle ABC with the given parts. 5. b = 20.7 ft., A = 34o30’, C = 102o50’ A) 349.2 ft2

B) 354.2 ft2

6. a = 46 ft., b = 59 ft., c = 69 ft.

7. a = 154 cm, b = 179 cm, c = 183 cm

C) 174.6 ft2

D) 169.6 ft2


Part III. Solve the problem. 8. Find the area of a triangular-shaped field with sides of 147.0 m and 154.9 m and the included angle between them measuring 60.94o. A) 6546 m2

B) 11,780 m2

C) 23,560 m2

D) 13,091 m2

9. Points A and B are on opposite sides of a lake. A point C is 84.5 meters from A. The measure of angle A is 79o20’, and the measure of angle C is determined to be 33o10’. Find the distance between points A and B (to the nearest meter). A) 23 m

B) 25 m

C) 48 m

D) 50 m

10. Two airplanes leave an airport at the same time, one going northwest at 414 mph and the other going east at 345 mph. How far apart are the planes after 4 hours (to the nearest mile)?


Trigonometry/Pre-Calculus Solving Oblique Triangles – Post-Test

Name: ______________________

Part I. Find the missing parts of the triangle. 1. C = 35o, a = 18 m, b = 22 m A) c = 17 m, A = 45o, B = 100o

B) c = 14 m, A = 49o, B = 96o

C) c = 13 m, A = 55o, B = 90o

D) c = 20 m, A = 30o, B = 115o

2. B = 25o, C = 40o, c = 40 A) A = 115o, a = 56, b = 26

B) A = 115o, a = 26, b = 56

C) A = 115o, a = 48, b = 36

D) A = 115o, a = 52, b = 42

3. B = 117.32o, a = 15.05 in., b = 67.25 in.

4. a = 24 yds., b = 32 yds., c = 28 yds.

Part II. Find the area of triangle ABC with the given parts. 5. a = 60 ft., b = 44 ft., C = 44o A) 1833.9 ft2 6. a = 9.0 km, b = 7.0 km, c = 12 km

7. a = 24 cm, b = 30 cm, c = 36 cm

B) 916.9 ft2

C) 949.5 ft2

D) 1899.1 ft2


Part III. Solve the problem. 8. A painter needs to cover a triangular region 75 meters by 68 meters by 85 meters. A can of paint covers 75 square meters of area. How many cans (to the next higher number of cans) will be needed? A) 75 cans

B) 2,436 cans

C) 48 cans

D) 33 cans

9. The original portion of the Luxor Hotel in Las Vegas has the shape of a square pyramid. Each face of the pyramid is an isosceles triangle with a base of 646 feet and sides of length 576 feet. Assuming that the glass on the exterior of the Luxor Hotel costs $35 per square foot, determine the cost of the glass, to the nearest $10,000, for one of the triangular faces of the hotel. A) $150,000

B) $5,390,000

C) $40,000

D) $4,250,000

10. A developer owns a triangular lot at the intersection of two streets. The streets meet at an angle of 72o, and the lot has 300 feet of frontage along one street and 416 feet of frontage along the other street. Find the length of the third side of the lot.


Unit%6%Exam% % % % % % % % Name___________________________________% MULTIPLE(CHOICE.((Choose(the(one(alternative(that(best(completes(the(statement(or(answers(the(question.( Solve(the(triangle,(if(possible.(( ( 1)%% C = 35°30', a = 18.76, c = 16.15 %(aligned(with(question(3(on(preF(and(postFtest) A)%%A%=%102°055,%B%=%42°255,%b%=%17.52;% A5%=%6°555,%B5%=%137°355,%b5%=%26.19%% C)%%No%solution%%

B)%%A%=%42°255,%B%=%102°055,%b%=%25.19%% % % D)%%A%=%42°255,%B%=%102°055,%b%=%27.20;% A5%=%137°355,%B5%=%6°555,%b5%=%3.35%

% % Solve(the(problem.( 2)%%Find%the%area%of%a%triangular-shaped%field%with%sides%of%174.0%m%and%154.9%m,%and%the%included%angle%between% them%measuring% 60.94° .%%(aligned(with(question(5(on(preF(and(postFtest) A)%%6546%m2%% B)%%11,780%m2%% C)%%23,560%m2%% D)%%13,091%m2% % % 3)%%Points%A%and%B%are%on%opposite%sides%of%a%lake.%A%point%C%is%84.5%meters%from%A.%%The%measure%of%angle%BAC%is%% 79°20 ' ,%and%the%measure%of%angle%ACB%is%determined%to%be%% 33°10 ' .%%Find%the%distance%between%points%A% and%B%(to%the%nearest%meter).%%(aligned(with(question(9(on(preFtest/question(2(on(postFtest) A)%%23%m%% B)%%25%m%% C)%%48%m%% D)%%50%m% % % Find(the(area(of(triangle(ABC(with(the(given(parts.( 4)%% b = 20.7 ft , A = 34°30', C = 102°50' %((aligned(with(question(5(on(preF(and(postFtest) A)%%349.2%ft2%%

B)%%354.2%ft2%%

C)%%174.6%ft2%%

D)%%169.6%ft2%

% % Determine(the(number(of(triangles(ABC(possible(with(the(given(parts.( 5)%% a = 24, b = 19, A = 44° %(aligned(with(question(3(on(preF(and(postFtest) A)%%3%% % % 6)%% b = 24, c = 28,

B)%%0%%

C)%%1%%

D)%%2%

B = 47° %(aligned(with(question(3(on(preF(and(postFtest)

A)%%1%% B)%%3%% C)%%2%% D)%%0% % % Find(the(missing(parts(of(the(triangle.( 7)%% C = 122.6°, a = 8.00 m, b = 8.16 m %%(aligned(with(question(3(on(preF(and(postFtest) A)%%No%triangle%satisfies%the%given%conditions.%% C)%%c%=%%14.2%m,%A%=%%28.3°,%B%=%%29.1°%% % % 8)%% C

= 109°, a = 44 km, b = 54 km %%(aligned(with(question(3(on(preF(and(postFtest) A)%%A%=%%45°,%B%=%81°,%b%=%65%km%% C)%%No%solution%%

% %% % %

B)%%c%=%%17.1%m,%A%=%%30.3°,%B%=%%27.1°% D)%%c%=%%20%m,%A%=%%26.3°,%B%=%%31.1°%

B)%%A%=%%44°,%B%=%82°,%b%=%62%km% D)%%A%=%%44°,%B%=%82°,%b%=%59%km%


9)%%(aligned(with(question(3(on(preF(and(postFtest)% % % 30o A % % % A)%% B = 60°, C = 90°, b = 3 C)%% B

= 90°, C = 60°, b = 3 3 %%

C 3 B

6 B)%%B

= 60°, C = 90°, b = 3 3 %

D)%%No%solution%

% Find(the(indicated(angle(or(side.( B 10)%%Find%the%measure%of%angle%A.% 2 13 (aligned(with(question(4(on(preF(and(postFtest)% % % 2 % % C A 6 % % A)%%135°%% B)%%140°%% C)%%60°%% D)%%120°% % % SHORT(ANSWER.((Write(the(word(or(phrase(that(best(completes(each(statement(or(answers(the(question.( Find(the(missing(parts(of(the(triangle.( 11)%% A = 65.3°, a = 2.15 km, b = 2.25 km %(aligned(with(question(1(on(preF(and(postFtest) % % % % % % %% % Find(the(missing(parts(of(the(triangle.((Find(angles(to(the(nearest(hundredth(of(a(degree.)( 12)%% a = 163 yd , b = 182 yd , c = 328 yd %(aligned(with(question(4(on(preF(and(postFtest) % % % % % %% % Solve(the(problem.( 13)%%Two%airplanes%leave%an%airport%at%the%same%time,%one%going%northwest%at%414%mph%and%the%other%going%east%at% 345%mph.%%How%far%apart%are%the%planes%after%4%hours%(to%the%nearest%mile)?%(aligned(with(questions(8(and(10( on(preF(and(postFtest) % % ( ( ( ( Find(the(area(of(triangle(ABC(with(the(given(parts.( 14)%% a = 46 ft , b = 59 ft c = 69 ft %(aligned(with(questions(6(and(7(on(preF(and(postFtest)


Trigonometry/Pre-Calculus The Law of Sines

Name: ______________________

Use the Law of Sines to solve each of the following oblique triangles. In all side-side-angle cases, there is exactly one triangle resulting from the given measures. 1. B = 52o, C = 29o, a = 43 cm

2. B = 18.7o, c = 124.1o, b = 94.6 m

3. C = 74.08o, B = 69.38o, c = 45.38 m

4. B = 38o40’, a = 19.7 cm, C = 91o40’

5. A = 35.3o, B = 52.8o, b = 675 ft.

6. C = 71.83o, B = 42.57o, a = 2.614 cm

7. A = 42.5o, a = 15.6 ft., b = 8.14 ft.

8. A = 96.80o, b = 3.589 ft., a = 5.818 ft.


Trigonometry/Pre-Calculus The Law of Sines – Homework

Name: ______________________

Directions: Each of the exercises has its answers contained in a box below. After solving each oblique triangle, cross out the box that contains the answers. The remaining box contains the answer to today’s trivia question: Who was the only president to graduate from the U.S. Naval Academy? 1. A = 50o, B = 70o, b = 15

2. A = 42o, B = 37o, a = 20

3. B = 25o, C = 68o, c = 10

4. A = 35o, B = 85o, b = 16

5. A = 54o, B = 25o, a = 30

6. A = 140o, a = 40, b = 26.3

7. C = 73o, b = 80.1, c = 100

8. A = 55o, B = 65o, a = 60

DWIGHT D. EISENHOWER

HARRY S. TRUMAN

FRANKLIN D. ROOSEVELT

C = 101o B = 50.28o c = 50.66

C = 60o a = 9.21 c = 13.91

C = 101o b = 15.67 c = 36.40

LYNDON B. JOHNSON

ULYSSES S. GRANT

JIMMY CARTER

C = 60o b = 66.38 c = 63.43

C = 60o a = 12.23 c = 13.82

C = 60o a = 9.21 b = 14.50

WOODROW WILSON

HERBERT HOOVER

WILLIAM HOWARD TAFT

B = 25o C = 15o c = 16.11

A = 57o B = 50o a = 87.70

A = 87o a = 10.77 b = 4.56


Trigonometry/Pre-Calculus The Law of Cosines

Name: ______________________

Use the Law of Cosines to solve each of the following oblique triangles. 1. A = 41.4o, b = 2.78, c = 3.92

2. A = 67.3o, b = 37.9, c = 40.8

3. C = 72o40’, a = 327, b = 251

4. A = 80o40’, b = 143, c = 89.6

5. B = 168.2o, a = 15.1, c = 19.2

6. a = 187, b = 214, c = 325

7. a = 4.0, b = 5.0, c = 8.0

8. a = 28, b = 47, c = 58


Trigonometry/Pre-Calculus The Law of Cosines – Homework

Name: ______________________

Directions: Each of the exercises has its answers contained in a box below, along with a letter. After solving each oblique triangle, find the box that contains the answers and place that letter in the space corresponding to the problem number (for example, if box M contains the answer to question 1, place an “M” in each blank with a “1” below it). The first letter has been given as a freebie. If properly filled in, the letters will spell out the answer to today’s trivia question: Who is the only president to never marry? 1. C = 28.3o, b = 5.71, a = 4.21

2. C = 45.6o, b = 8.94, a = 7.23

3. B = 74.80o, a = 8.919, c = 6.427

4. C = 59.70o, a = 3.725, b = 4.698

5. A = 112.8o, b = 6.28, c = 12.2

6. a = 3.0, b = 5.0, c = 6.0

7. a = 9.3, b = 5.7, c = 8.2

8. a = 42.9, b = 37.6, c = 62.7

9. c = 1240, b = 876, a = 918 B

C

A

A = 30o B = 56o C = 94o

A = 82o B = 37o C = 61o

A = 64.59o C = 40.61o b = 9.529

S

U

N

A = 44.9o B = 106.8o c = 2.83

A = 48.75o B = 71.55o c = 4.276

A = 47.7o B = 44.9o C = 87.4o

M

E

H

A = 53.1o B = 81.3o c = 6.46

A = 42.0o B = 35.9o C = 102.1o

B = 21.6o C = 45.6o a = 15.7

_J_ ___ ___ ___ ___ (3) (2) (8) (1)

___ ___ ___ ___ ___ ___ ___ ___ (6) (4) (7) (5) (3) (9) (3) (9)


Trigonometry/Pre-Calculus Mixed Practice: The Laws of Sines and Cosines

Name: ______________________

Solve each of the following oblique triangles. Remember that you can determine whether to use the Law of Sines or Law of Cosines by the setup of the triangle. 1. a = 4.0, b = 5.0, c = 8.0

2. a = 28, b = 47, c = 58

3. B = 59o, a = 10, c = 9

4. A = 71o, C = 59o, b = 17

5. B = 80o, a = 32, b = 36

6. A = 92o, B = 33o, a = 22

7. A = 30o, C = 111o, b = 29

8. a = 23, b = 21, c = 28


9. a = 14, b = 28, c = 17

10. C = 114o, a = 27, b = 23

11. B = 79o, a = 25, c = 18

12. a = 18, b = 24, c = 26

13. a = 12, b = 28, c = 18

14. C = 75o, a = 16, b = 11

15. a = 23, b = 30, c = 13

16. A = 10o, C = 140o, a = 6

17. A = 42o, B = 53o, b = 16

18. B = 115o, b = 18, c = 12


Trigonometry/Pre-Calculus History of Mathematics Readings

Name: ____________________

Topic:

Big Idea/Overarching Theme

Give an example.

Give an example.

Give an example.

What, if any, were the effects of this example at the time?

What, if any, were the effects of this example at the time?

What, if any, were the effects of this example at the time?

How has this example affected the material we have studied thus far?

How has this example affected the material we have studied thus far?

How has this example affected the material we have studied thus far?

Other thoughts?

Other thoughts?

Other thoughts?


Trigonometry/Pre-Calculus Finding Areas of Oblique Triangles – Homework

Name: ______________________

Find the area for each of the following oblique triangles. Sorry, no presidential trivia today. 1. C = 72.2o, a = 35.1 ft., b = 43.8 ft.

2. C = 142.7o, a = 21.9 km, b = 24.6 km

3. A = 56.80o, b = 32.67 in., c = 52.89 in.

4. A = 34.97o, b = 35.29 m, c = 28.67 m

5. A = 24.67o, C = 52.67o, b = 27.3 cm

6. a = 22 in., b = 45 in., c = 31 in.

7. a = 154 cm, b = 179 cm, c = 183 cm

8. a = 25.4 yd., b = 38.2 yd., c = 19.8 yd.

9. a = 76.3 ft., b = 109 ft., c = 98.8 ft.

10. a = 15.89 in., b = 21.74 in., c = 10.92 in.


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 are only three minorities 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, three received grades of “A” in their most recent math class, five received a “B,” and one received a “C,” giving it an ideal mix. Members of group: A., B., C., F., H., I., L., M., N. Students Below 90 Percent Attendance Consistent attendance is crucial to high achievement in high school, particularly in a sequential course such as this one, in which each day’s material builds on what was introduced in the previous class. This is particularly true in a school year such as this one, which has featured 12 snow days to date (in addition to eight two-hour delays). Ninety percent is generally the benchmark used for the minimum acceptable attendance level and will thus be used as the cutoff for this subgroup as well; entering this unit, six students had below 90 percent attendance this semester. (For the purposes of identifying this subgroup, lateness in excess of 30 minutes is counted as an absence due to the substantial instructional time missed.) This unit requires indepth knowledge of the material and techniques introduced in the preceding unit; although students can of course request extra help to cover material they missed (and are expected to do so following absences), I wish to examine this subgroup to determine how effective these strategies are at providing students with sufficient content knowledge, and how crucial strong attendance truly is to success in mathematics. Members of group: C., F., G., H., M., S.


Section 4: Lesson Plans


Pretest Item Analysis (Whole Class) Grade: 11/12 Subject: Trigonometry/Pre-Calculus Objectives:

Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5

Unit: Solving Oblique Triangles

Intern: Ben Cohen

Use the Law of Sines to solve oblique triangles given AAS, ASA, or SSA. Use the Law of Cosines to solve oblique triangles given two sides and the included angle (SAS). Use the Law of Cosines to solve oblique triangles given all three side measures (SSS). Find the area of an oblique triangle using Heron’s Formula given SSS. Find the area of an oblique triangle given SAS, ASA, or AAS; solve application problems. PRETEST ASSESSMENT ITEMS

1

2

3

STUDENT

2 3

5

6

7

8

9

10

5

1

2

ALIGNMENT WITH LESSON OBJECTIVES

2 1

4

X X X

1

1

3

X X

5

4

4

X

X

4

X

5 6

X

X

7

X X X

8 9 10 11 12

X X

13 14 15 16 TOTAL CORRECT RESPONSES

X X 8

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

X X X X X X 8

X X X X

X 0

0

6

0

0

4

2

0

TOTAL CORRECT RESPONSES

1 2 4 0 1 2 0 2 2 2 2 3 2 2 1 2 28


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

Students were most successful at solving multiple-choice problems given either sideangle side (SAS) or side-side-angle (SSA); eight out of 16 students (50 percent) correctly solved each of these problems, including five who correctly solved both problems. To an extent, this is a surprisingly high percentage since students have not yet explicitly learned how to solve these problems; however, since both methods are derived from material with which students are already familiar (such as the Pythagorean theorem), it is possible these students were able to extend their knowledge, aided by having the answer choices provided. However, no students were able to solve a free-response SSA or SAS problem, and only 25 percent (four out of 16) were able to solve a word problem requiring the use of SAS. Thus, it appears that a number of the correct answers may simply be attributable to successful guessing; at the same time, this does suggest that the derivation of these formulas, which I was unsure of whether to originally include, is in fact not too advanced for this class and should be included in the lessons.

No students were able to solve an oblique triangle given all three side measures. While this technique is also closely related to the Pythagorean theorem, it is not derived from it, per se; rather, the Pythagorean theorem is a special case of the Law of Cosines, which is used to solve this problem. As such, students cannot be reasonably expected to translate their existing knowledge of the Pythagorean theorem into solving a problem requiring the Law of Cosines; this will be introduced as entirely new material, although the connections between the two formulae will be highlighted.

Similarly, no students were able to solve either of the two problems that asked them to use Heron’s formula to find the area of an oblique triangle. Again, this is not surprising since students have had no prior exposure to this, and the derivation of this formula does not immediately follow from students’ existing knowledge. Like with the Law of Cosines, this will be treated as entirely new material, but the derivation will be included.

Students did surprisingly well at finding the area of an oblique triangle using the sine formula, with six out of 16 (37.5 percent) correctly answering question 5 and four out of 16 (25 percent) correctly answering question 8. However, it appears the success on question 5 is attributable to students’ guessing strategy: Some students informed me following the administration of the test that they had learned to guess “C” on a multiplechoice question if they do not know the answer; as it happens, that was the answer to this question. In addition, the 25 percent success rate on question 8 is exactly what would be expected if all students guessed randomly. Thus, the relatively high rate of correct responses is presumably not due to students’ existing knowledge.

Only two students correctly answered question 9, and none correctly answered question 10; these two questions were word problems strongly correlated with questions 2 and 1, respectively, which received eight correct answers each. This suggests I will need to devote instructional time to solving these problems, rather than expecting students to be able to instantly apply their knowledge of straightforward calculations to word problems.


Pretest Item Analysis Disaggregated Subgroup – Female Students Grade: 11/12 Subject: Trigonometry/Pre-Calculus Objectives:

Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5

Unit: Solving Oblique Triangles

Intern: Ben Cohen

Use the Law of Sines to solve oblique triangles given AAS, ASA, or SSA. Use the Law of Cosines to solve oblique triangles given two sides and the included angle (SAS). Use the Law of Cosines to solve oblique triangles given all three side measures (SSS). Find the area of an oblique triangle using Heron’s Formula given SSS. Find the area of an oblique triangle given SAS, ASA, or AAS; solve application problems. PRETEST ASSESSMENT ITEMS

1

2

3

2

1

1

X

X

STUDENT 1

4

5

6

7

8

9

10

5

1

2

ALIGNMENT WITH LESSON OBJECTIVES

3

5

4

4

2

X

3 4

X

X X X

5 6 7 8

X X

9 TOTAL CORRECT RESPONSES

4

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

X X X 4

X X X

0

0

X 4

0

0

3

1

0

TOTAL CORRECT RESPONSES

2 0 1 2 2 2 2 3 2 16


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 using the Law of Sines and the SAS case of the Law of Cosines; four out of nine students (44 percent)—a rate nearly on par with the 50 percent performance of the class as a whole—were able to successfully solve these problems. In addition, three students (C., L., and I.) correctly answered both problems. As with the class as a whole, however, no students were able to correctly answer free-response questions concerning these techniques. Much with the class as a whole, then, it appears that these students are able to derive these formulas from their pre-existing knowledge at a basic level and ballpark the answers but cannot entirely solve such problems on their own without assistance. It is refreshing to see that the female students can perform such derivations just as effectively as their male peers; as was stated in the full-class analysis, these derivations will be included in the lessons. (It is also worth noting that one of only two students to get more than two problems right—I., who correctly answered three questions—is a member of this subgroup.)

No female students were able to solve an SSS oblique triangle. Much like with the full class (which also saw no students able to solve such a triangle), this is likely attributable to the complexity of the derivation of this formula—rather than being derived from the Pythagorean theorem, this is a special case of the Pythagorean theorem, and so it is significantly easier to start with the Law of Cosines and re-derive the Pythagorean theorem than to go in reverse. While I will be sure to highlight the close connections between the two formulas, this will be treated as being entirely new to students.

Another side-side-side formula that students were unable to solve questions related to is Heron’s formula; questions 6 and 7 (both free-response) both received no correct responses from the female subgroup. Again, this is a very complicated formula to derive, and students cannot be reasonably expected to arrive at it based solely on their prior knowledge; I will presuppose no such prior knowledge when introducing it.

Four out of nine students (44 percent) correctly answered question 5, with three (33 percent) correctly answering question 8; these two questions asked students to find the area of an oblique triangle using the sine formula. As with the whole class, however, it appears students’ success on question 5 was due to the strategy of guessing “C” on multiple-choice questions, and the results on question 8 are again what would be expected from random guessing. As such, it would be improper to make any inferences regarding students’ existing knowledge from these results.

Only one student (F.) correctly answered question 9, and none correctly answered question 10; these two questions were word problems that asked students to use the same techniques as questions 2 and 1, which each received four correct responses. (The only student to correctly answer question 9 incorrectly answered both questions 1 and 2.) Again, this suggests I will need to demonstrate how to apply techniques learned previously in the unit to solve word problems.


Pretest Item Analysis Disaggregated Subgroup – Students Below 90% Attendance Grade: 11/12 Subject: Trigonometry/Pre-Calculus Objectives:

Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5

Unit: Solving Oblique Triangles

Intern: Ben Cohen

Use the Law of Sines to solve oblique triangles given AAS, ASA, or SSA. Use the Law of Cosines to solve oblique triangles given two sides and the included angle (SAS). Use the Law of Cosines to solve oblique triangles given all three side measures (SSS). Find the area of an oblique triangle using Heron’s Formula given SSS. Find the area of an oblique triangle given SAS, ASA, or AAS; solve application problems. PRETEST ASSESSMENT ITEMS

1

2

3

2

1

1

X X

X

STUDENT 1 2

4

5

6

7

8

9

10

5

1

2

ALIGNMENT WITH LESSON OBJECTIVES

3

5

4

4

3

X

4

X

5 6 TOTAL CORRECT RESPONSES

X 3

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

X 2

0

0

1

X 0

0

1

1

0

TOTAL CORRECT RESPONSES

1 2 0 1 2 2 8


Instructional Implications Based on Analysis of Pretest Data (Disaggregated Subgroup: Students Below 90% Attendance) •

The most striking takeaway from this subgroup is that it includes three of the five students who received the lowest scores on the pretest (G., H., and F.), including one of only two zeros. In addition, no student in this subgroup correctly answered more than two questions. While some of this can be explained by random variation, the students’ lower levels of performance relative to their peers certainly lend credence to the theory that absences significantly impact student learning.

Like the full class and the female students only, this subgroup was most successful at using the Law of Sines and the SAS case of the Law of Cosines; these questions (1 and 2) were the only ones that received more than one correct response. Two of the three students who correctly answered question 1 (SAS) also correctly answered question 2 (Law of Sines): C. and S. The rates of correct responses to these two questions were roughly on par with the full class; this is likely a consequence of these questions extending material to which students were exposed several years ago in Geometry, rather than earlier in the semester.

No students in this subgroup were able to solve an oblique triangle given all three sides, which exactly paralleled the results of the full class. Again, this is likely attributable to the complexity of the derivation of this formula—rather than being derived from the Pythagorean theorem, this is a special case of the Pythagorean theorem, and so it is significantly easier to start with the Law of Cosines and re-derive the Pythagorean theorem than to go in reverse. This data point cannot thus be blamed on students’ attendance, but rather the inherent difficulty of the question.

No students in this subgroup correctly answered questions 6 and 7, which asked students to use Heron’s formula to find the area of an oblique triangle given SSS. Like with the SSS case of the Law of Cosines, the derivation of this formula is exceptionally complicated, and students at this level cannot be reasonably expected to determine it with no prior knowledge.

Questions 5 and 8 each received only one correct answer, both from the same student (M.)—who answered only these two questions correctly. Incidentally, this student has the lowest attendance in the class (50 percent at the start of this unit); it is unclear if she was using the “guess ‘C’” strategy (both questions were multiple-choice) or if the data contradict the assumptions about the correlation between attendance and performance.

The two word problems (questions 9 and 10) received only one correct answer between them (F. correctly answered question 9, though she incorrectly answered both questions 1 and 2). This would seem to suggest on the surface that these students’ problem-solving abilities are less well developed than those of their peers, although considering the low number of correct responses among the whole class (two to question 9 and none to question 10), this may not be an accurate inference.


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 February 24, 2014 Mentor Mark Heffner Standard(s): ____ CCSS

Unit Solving Oblique Triangles Grade Class Size 11/12 16 School Winters Mill High School

Topic Law of Sines Time 7:30-8:50 a.m. Intern Ben Cohen

Other: Maryland Content Standards

2.2.2.1 The student will apply the Law of Sines and the Law of Cosines to solve problems involving oblique triangles. Big Idea or Essential Question In the previous unit, students learned how to use trigonometric functions to solve right triangles. This unit extends this knowledge by applying trigonometric functions to oblique (i.e., non-right) triangles. Students will learn how to use two fundamental identities—the Law of Sines and the Law of Cosines—to solve these triangles given a series of side lengths or angle measures. Today’s lesson will focus on the former—and simpler—of these two laws. Throughout this unit, students will also be exposed to the myriad cross-curricular connections in which these laws are used, such as determining the best way to play a golf hole or measuring the distance of a storm from the coast; like in the last unit, this will be a gradual process of exposure.

Alignment with Summative Assessment Questions 1, 9, and 11 on the CCPS summative assessment ask students to use the Law of Sines to solve an oblique triangle. Question 9 includes a sketch of the triangle, while questions 1 and 11 include only the respective side and angle lengths. Lesson Objective Given two angle measures and one side length or two side lengths and the non-included angle in an oblique triangle, students will use the Law of Sines to solve the triangle. Formative Assessment Students will be asked to complete a series of problems for classwork and homework. During class, they will be asked to complete a handout with eight problems, all of which will ask them to use the Law of Sines to solve oblique triangles. For homework, they will complete an additional eight problems similar to those covered in class. Informal assessments are integrated throughout the lesson: the Geometer’s Sketchpad demonstration will serve as an informal assessment technique that allows me to measure how much material students have retained from their ninth-grade geometry classes by observing both what percentage of students volunteer to answer questions related to the demonstration and whether their answers are correct. Similarly, the algebraic manipulations needed to complete this derivation are also required to solve problems using the Law of Sines; I will informally assess students’ comfort level with these manipulations to see if any further practice or remediation is necessary.


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.

• Fifteen out of 16 students received a 32/35 (91 percent) or higher on Friday’s quiz on solving right triangles. Five students scored 100 percent.

• This suggests that the class has fully mastered the techniques of solving right triangles. Although the techniques used for oblique triangles differ from those for right triangles, it is essential that students have solid mastery of right-triangle solving since oblique-triangle solving builds on these techniques. Given this mastery, I can move directly into techniques for oblique triangles without having to review solving right triangles first.

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

• Students were able to grasp the procedure for solving a right triangle more quickly than I had anticipated—they did not need as many examples as I had planned.

Individual or Small Group Needs Specific to this Lesson

• I will be ready for them to grasp this material similarly quickly, particularly since the Law of Sines is the simpler of the two laws. However, it is still crucial that I have extra examples prepared in case the students take longer than I anticipate so I do not end up short-changing them or give them insufficient practice.

►►►►►

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

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

• M. was absent for the final three days of the preceding unit on right triangles. Between illness and snow days, she has been in class just four days this month.

• Since she missed almost the entire right-triangle solving unit, it is likely she will need significant remediation. However, while this unit builds on the techniques used in the previous one, it is likely she will be able to catch on to this unit in the early lessons, allowing me to use time out of class rather than instructional time to catch her up.

• Care must be taken to engage all learners, being mindful of the gender split and high percentage of FARMS students.

• As is done in all lessons, any real-world examples will be as universally applicable as possible and will avoid making reference to luxuries that some students may not have experienced. Also, any discussions of calculator techniques must be carefully framed—rather than asking which students own a TI-83 or TI-84, for example, I will simply preface this with “If you have a TI-83 or a TI84…”


III. Instructional Procedures – What instructional strategies will I use to ensure that every child is a successful learner? Instructional Materials/Resources The instructor will prepare a handout with eight problems on it for classwork as well as eight additional problems for homework. All problems will ask students to solve a given oblique triangle using the Law of Sines and will make use of ASA, SAA, and SSA scenarios. This lesson will also make extensive use of the TI-Nspire calculators located in the classroom as well as a demonstration in Geometer’s Sketchpad to derive the law.

Technology Integration When solving oblique triangles, extensive use will be made of the TI-Nspire calculators in the classroom. This is particularly necessary for this class, which in some cases will require students to go backwards and find the angle corresponding to a pair of given side lengths. The lesson will also include a component describing for students how to perform these calculations on a TI-83 or TI-84. In addition, a demonstration in Geometer’s Sketchpad will be used to help students derive the Law of Sines. This will also serve as an informal assessment technique; I will pose questions throughout the demonstration (e.g., “Who remembers what is always true about the angle measures in an equilateral triangle?”) to see how much material students have retained from Geometry and will note both how many students volunteer to answer the questions and the accuracy of their responses. Cross-curricular Connections Students will be exposed to a variety of applications to other fields that require the use of the Law of Sines. Examples will be drawn from various fields, including architecture (finding the height of a building or broadcast antenna), meteorology (finding the distance from the U.S. coastline of a storm at sea), maritime navigation (finding the distance between two ships and land), and sports (determining the best way to play a golf hole). Students will be exposed to these applications, but will not be expected to solve such problems until Wednesday’s lesson, which will introduce them to the techniques for solving such problems. 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 equations 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. • As the opening lesson of the unit, this lesson is slightly more lecture-heavy than those that follow it. To ensure students do not lose interest and remain invested in the class, opportunities for student participation are included throughout the lecture; the instructor will circulate during the individual-work portion of the class as well to continue engaging 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 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 (finding unknown side length and angle measures in an oblique triangle). • Following this, student knowledge will again be verified by a warm-up activity that assesses students’ calculator skills and ability to find the values of sine and cosine for nonstandard acute angles. Although it may appear trivial, their ability to use the classroom’s TI-Nspire calculators to evaluate these functions is essential to success in this unit.

Development of the New Learning

30 minutes

(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.)

• Begin by using the example of my trip to Alaska from several years ago: I flew in to Fairbanks and then traveled on land to Seward and by sea to Vancouver. Unlike in our previous examples, however, the line segments connecting these three cities do not form a right angle. How, then, can we find the distance connecting Fairbanks to Vancouver, the longest side of the triangle? • Use a demonstration in Geometer’s Sketchpad to show students how to solve a similar angle-side-angle example. We can construct a perpendicular line from one vertex to the opposite side and create two right triangles, and then use the definition of sine, which was covered extensively in the previous unit. After algebraically rearranging the terms, we arrive at the Law of Sines:

a b c . = = sin A sin B sinC • Following this derivation, use the Law of Sines to solve the original example: the distance from Fairbanks to Vancouver is approximately 1,402 miles. Enrichment or Remediation (As appropriate to lesson)

30 minutes

• Give students another example: Suppose we know the length of side a of some triangle and the measures of angles B and C. o Since we know two angle measures, we can find the third (180 – B – C); once we have this information, we can again substitute into the equation for the Law of Sines to find the lengths of sides b and c. • The final example is the case in which two sides and the nonincluded angle are known. Go through two examples of this—one in which the angle is acute and one in which it is obtuse. In both cases, there will be exactly one triangle that can be formed from the given sides and angle. However, inform students that it is possible in some cases that there can be two triangles formed, with the second triangle including the supplement of the given angle. This is known as the ambiguous case and will be covered in a later lesson. • For classwork, students will complete eight problems asking


them to solve an oblique triangle using the Law of Sines. Questions will be roughly evenly split between ASA, SAA, and SSA. • Circulate as students are working to answer questions and ensure all students are staying on task. • When students have finished, if there is enough time remaining, have one student put each problem up on the board so the class can discuss them. It is expected there will be sufficient time remaining to facilitate this. Assessment/ Evaluation

Ongoing

• Students will be informally assessed throughout the lesson. The Geometer’s Sketchpad demonstration will serve as an informal assessment technique; I will use it to get an idea of how much knowledge students have retained from ninth-grade Geometry (I will observe both how many students volunteer answers and whether their answers are correct). 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 during and after the lesson to ask if any students have questions about the material. • As an additional formative assessment, students will solve an additional eight oblique triangles, again using the Law of Sines. As was done previously, questions will be roughly evenly split between ASA, SAA, and SSA.

Planned Ending (Closure) • •

5 minutes

Summary Homework

• 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 eight problems).

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?

• Although the Geometer’s Sketchpad demonstration appeared to be effective, the derivation of the Law of Sines using GSP took longer than I had expected. As a consequence, students were only able to complete one of the formative assessments (the classwork and homework were combined). I am also planning to use a demonstration in Geometer’s Sketchpad to illustrate the derivation of the Law of Cosines, but I will reevaluate this and move to a simplified derivation if it appears the Sketchpad demonstration will take too long. • Twenty-five percent of students (four out of 16) encountered difficulty with the algebraic manipulations necessary to solve a


proportion in a Law of Sines-related problem. While this is certainly somewhat distressing, as this knowledge is presupposed for students in this class, this is nevertheless a problem that is easily rectified—since they have already covered how to solve proportions in Intermediate Algebra and Algebra II, they likely simply need a quick refresher. When going over the homework tomorrow and also when solving Law of Cosines problems, I will be sure to go through all algebraic steps to assess students’ familiarity and comfort with these derivations; if they are comfortable with solving these problems, I can then speed up the demonstrations and begin skipping steps. I now realize it is much easier to go from showing all steps to showing fewer ones than in the opposite direction; I will use this approach in subsequent lessons (in both this and future units). • Algebra difficulties aside, it is refreshing to see that students were able to quickly grasp the mathematics behind the Law of Sines; they experienced near-universal success when using it to solve a triangle (with the most frequent mistakes being simple arithmetic errors). It appears recall of material from Geometry will be a strength for this class during this unit; I am eager to see if this trend continues later in the unit. 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 stated earlier, the Geometer’s Sketchpad demonstration— though it took longer than I had planned—turned out to be very helpful for students. I was initially concerned about their ability to recall material from Geometry—and even earlier in this course—that was crucial to the GSP derivation, but the students encountered no problems with this. I do have a similar demonstration for the Law of Cosines, though if it requires too much instructional time it may be more of a hindrance than a help; as such, I plan to reassess the use of the demonstration when planning for the lesson tonight. • Although all students have taken Intermediate Algebra and Algebra II, the comfort level of some students with algebraic manipulations and derivations is still below what it should be. That said, it occurred to me that it is significantly easier (and more beneficial to students) to begin a lesson by showing all steps in the manipulation of an equation or proportion and then condensing the demonstrations only if students are comfortable enough, rather than beginning by skipping steps and only showing them upon students’ requests. I will certainly stick to the former method in subsequent lessons both within this unit and beyond it; while it may be acceptable to skip steps in a level 8 (honors) class, I realize I may have been asking too much from these students, even if they are expected to know how to perform these manipulations. • Ordinarily, this and part of the following lesson would be devoted to the ambiguous case of the Law of Sines (that is, those in which two sides and a non-included angle can lead to two separate triangles, or even no possible triangles). Because of compression of this unit due to snow, however, these cases will not be covered in the detail I would like. I do hope to give them more than a token mention in a subsequent class, but I will not be able to cover them to


the ideal extent because of this schedule compression.

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 an advanced, high-achieving class, there are still some students who occasionally struggle with basic concepts such as the values of special acute angles and the definitions of sine and cosine; 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 and are constantly pushing themselves. This lesson has been designed with both of these groups of students in mind, with activities designed to both introduce the new material and simultaneously activate students’ knowledge of the prerequisite skills they will need to be successful and demonstrations and derivations that provide both another way of looking at the material for the lower-achieving students and a challenge for the higher-achieving ones. The heterogeneous mix of student abilities will likely be reflected in the summative assessment for this unit (as in past units), but this lesson, as is always my goal, is designed to set up students all along the ability spectrum for success in the unit. • InTASC 6 (Assessment to Prove and Improve Student Learning): In a unit-opening lesson such as this one, assessment of students’ existing knowledge and comfort level with the prerequisite skills is essential. Arguably just as important is differentiation of these assessments in lieu of simply giving a pretest at the start of each unit. While this lesson did include a pretest, it also included numerous additional assessment techniques that were just as important. For example, the Geometer’s Sketchpad demonstration, in addition to providing a vehicle for the derivation of the Law of Sines, also allowed me to assess students’ existing knowledge of triangles and the extent to which they retained knowledge from their ninth-grade geometry classes by observing both the frequency with which students volunteered to answer questions and the accuracy of their answers. In addition, although it progressed in a manner opposite what I should have done, the algebraic derivations of the Law of Sines provided another assessment technique for me, allowing me to again assess students’ comfort level with the manipulations necessary—which are also required in order to solve problems—and determine whether any remediation or additional practice was necessary. • COE 11 (Use of Technology): The demonstration in Geometer’s Sketchpad was a focal point of this lesson, but merely integrating technology for technology’s sake is insufficient; it must be implemented in a way that will allow students to reinforce and extend their knowledge rather than a way that allows them to simply use it as a crutch. I felt that it was essential to show students how the Law of Sines is derived, rather than simply contrasting it with methods for solving a right triangle; in addition, I wanted to have an interactive demonstration as I did this and one that could be modified as necessary to highlight the essential facets (such as shading in one triangle) instead of one that would simply be done at the board (albeit with student input). As such, I instantly thought of Sketchpad, which is exceptionally useful for demonstrations such as this one. (Another benefit is allowing students to instantly confirm the relevant side and angle measures that they have derived from existing knowledge.) Technology integration is always at the forefront of my mind when planning a lesson, and this one was no exception; rarely, however, is it possible for technology to be integrated as seamlessly and beneficially as it was in this lesson. In addition, as was described above, this provided a useful informal assessment tool for me: I could use the frequency and accuracy of students’ responses to my prompting questions (such as angle measures in an equilateral triangle) to assess both the extent to which they retained knowledge from their prior classes as well as their comfort level with the new material. Certainly, it must be noted that the variety of technology used must be varied; I cannot simply do a demonstration in Sketchpad every class. However, this particular demonstration perfectly fit into and complemented today’s lesson.


Acute Triangle

Obtuse Triangle


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 February 25, 2014 Mentor Mark Heffner Standard(s): ____ CCSS

Unit Solving Oblique Triangles Grade Class Size 11/12 16 School Winters Mill High School

Topic Law of Cosines Time 7:30-8:50 a.m. Intern Ben Cohen

Other: Maryland Content Standards

2.2.2.1 The student will apply the Law of Sines and the Law of Cosines to solve problems involving oblique triangles. Big Idea or Essential Question Today’s lesson focuses on the second—and more complicated—of the two techniques for solving oblique triangles, the Law of Cosines. Students will first be shown how to derive this law and then shown how to use it; however, unlike the Law of Sines, it is only necessary to use the Law of Cosines once to set up a situation in which it is possible to use the Law of Sines to complete the process and solve the triangle. In addition, students will be shown that the Pythagorean theorem is simply a special case of the Law of Cosines, when angle C is equal to 90 degrees. This lesson will again expose students to the myriad cross-curricular connections in which these laws are used; this will continue to be a gradual process of exposure, setting up students for success when they tackle these problems during the following lesson.

Alignment with Summative Assessment Questions 10 and 12 on the CCPS summative assessment ask students to solve an oblique triangle using the Law of Cosines. Question 10 includes a sketch of the triangle, while question 12 gives only the lengths of the three sides. Lesson Objective Given two sides and the included angle or three sides of an oblique triangle, students will use the Law of Cosines to solve the triangle. Formative Assessment Students will be asked to complete a series of problems for classwork and homework. During class, they will be asked to complete a handout with eight problems, all of which will ask them to use the Law of Cosines to solve oblique triangles. For homework, they will complete an additional nine problems similar to those covered in class. Informal assessments are integrated throughout the lesson; remediation of past material will be included as the new material is introduced if it becomes necessary.


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.

• Only four students correctly answered question 8 on the pretest; this question asked students to solve an oblique triangle using the Law of Cosines. None correctly answered questions 4 or 10, which were free-response questions asking the same.

• This shows students have no prior knowledge of the Law of Cosines and are unfamiliar with how to use it to solve oblique triangles. As expected, I will have to introduce this as entirely new material. (It is likely that the four correct responses to question 8 were the result of random guessing rather than pre-existing knowledge.)

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

• Yesterday’s derivation of the Law of Sines took longer than expected, cutting into time for examples and classwork.

• It would be inappropriate to include remediation on this for the whole class, as this makes use of concepts students at this level are expected to have previously mastered. However, I will need to be aware that there are some students who would benefit from having me work with them one-on-one for such remediation.

• Some students encountered difficulty with the algebraic manipulations necessary to solve a proportion and find a missing side or angle measure with the Law of Sines.

Individual or Small Group Needs Specific to this Lesson

• Particularly since the Law of Cosines is the more complicated of the two, I will need to be prepared for this contingency. This derivation is extremely rigorous and is almost certainly the most difficult calculation students will perform in this unit. Depending on how many students are having trouble with the algebra, it may be advisable to ask them to speak to me after the lecture.

►►►►►

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

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

• M. has now been absent for four consecutive days. Between illness and snow days, she has been in class just four days this month.

• Since she missed almost the entire right-triangle solving unit, it is likely she will need significant remediation. She also has a substantial amount of work to make up; although she likely will not need to make up all of the right-triangle unit work to experience success in this unit, it is possible the material to be made up will continue to snowball depending on how long she is ill.

• Care must be taken to engage all learners, being mindful of the gender split and high percentage of FARMS students.

• As is done in all lessons, any real-world examples will be as universally applicable as possible and will avoid making reference to luxuries that some


students may not have experienced. Also, any discussions of calculator techniques must be carefully framed—rather than asking which students own a TI-83 or TI-84, for example, I will simply preface this with “If you have a TI-83 or a TI84…”

III. Instructional Procedures – What instructional strategies will I use to ensure that every child is a successful learner? Instructional Materials/Resources The instructor will prepare a handout with eight problems on it for classwork as well as nine additional problems for homework. All problems will ask students to solve a given oblique triangle using the Law of Cosines and will make use of both SAS and SSS scenarios. This lesson will also make extensive use of the room’s TI-Nspire calculators as well as a demonstration in Geometer’s Sketchpad to derive the law.

Technology Integration When solving oblique triangles, extensive use will be made of the TI-Nspire calculators in the classroom. This is particularly necessary for this class, which in all cases will require students to go backwards and find the angle corresponding to a pair of given side lengths. The lesson will also include a component describing for students how to perform these calculations on a TI-83 or TI-84. In addition, a demonstration in Geometer’s Sketchpad will be used to help students derive the Law of Cosines, an extremely challenging derivation that I nonetheless feel students will be able to complete. Cross-curricular Connections Students will be exposed to a variety of applications to other fields that require the use of the Law of Cosines. As was done in the previous lesson, examples will be drawn from various fields, including architecture (finding the height of a building or broadcast antenna), meteorology (finding the distance from the U.S. coastline of a storm at sea), maritime navigation (finding the distance between two ships and land), and sports (determining the best way to play a golf hole). Students will be exposed to these applications as a way of making the content meaningful for them, rather than simply making it another topic in the curriculum that they feel has no relevance or applicability to their everyday lives. In addition, examples will be as universal as possible to ensure no learners are unnecessarily excluded. 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 equations 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. • The administration has hinted that there may be an emergency drill this week and asked all teachers to remind students of these procedures. As such, it is possible class may be interrupted for an extended period of time one day this week. • 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 5 minutes

Warm-up Motivation Bridge

Procedure • For the warm-up, students will be asked to solve an oblique triangle using the Law of Sines; they will be given two angles and the non-included side. This scenario will be used since it tests their ability to find both unknown side lengths and angle measures. • Today’s lesson will focus on the second method for solving oblique triangles: the Law of Cosines. This is used when three sides or two sides and the included angle are given.

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.)

40 minutes

• Begin with a sports example: Rickey Henderson lays down a bunt, and the ball travels 26 feet in front of home plate before the pitcher fields it. At that time, how far is the pitcher from first base? • This is a case different from those we studied yesterday: here, we are given two sides and the included angle. Since we do not know any side-angle pairings, however, we cannot use the Law of Sines. How can this triangle be solved? • Use a demonstration in Geometer’s Sketchpad to show students how to solve a similar side-angle-side example. We can set up an oblique triangle in the coordinate plane so that point B is at the origin and point C is on the x-axis. We can derive the coordinates of point A in terms of the sine and cosine of point B, and then use the distance formula to find the length of side b. After algebraically rearranging the terms, we arrive at one form of the Law of Cosines:

b 2 = a 2 + c 2 − 2ac cos B

• Explain to students that it can take three forms; in each case, the first squared side and the opposite angle must match up, with the other two sides being included on the right side of the equation. • Following this derivation, use the Law of Cosines to solve the original example: the pitcher is approximately 74 feet from first base at the time the bunt is fielded. • Give students another example: Suppose we know the length of sides a and b of some triangle and the measure of angle C. We must begin by substituting this information into the equation for the Law of Cosines to find the length of side c. Once this is known, we have a side-angle pairing, and we can use the Law of Sines to complete the process of solving the triangle. • Now, suppose all three sides of a triangle are known, but no angle measures are given. Go through two examples of this—one in which the angle is acute and one in which it is obtuse. Once again, we can choose one angle measure to find and use the Law of Cosines to do so; once this is done, we will have a side-angle pairing and can use the Law of Sines to determine the remaining angle measures.


Enrichment or Remediation

30 minutes

(As appropriate to lesson)

• At this point, discuss with students some of the applications of the Law of Sines and the Law of Cosines with an eye towards making these topics more meaningful and accessible. Possible applications include meteorology (e.g., finding the distance of Superstorm Sandy from the New England and New York coastlines), sports (in addition to the bunt example from the start of class, this can be used to find the distance needed for a pass in soccer or lacrosse, or whether an open-field tackle can be made in football), and navigation (using cell phone towers in conjunction with a GPS system to pinpoint the location of a 911 caller). • Ask students to think about an application that is particularly meaningful for them—for example, the JROTC students may find navigational examples (e.g., the U.S. Navy determining the respective distances of a pair of enemy submarines) particularly applicable. All students will be asked to share an example that they have come up with. • Following this discussion, students will complete for classwork eight problems asking them to solve an oblique triangle using the Law of Cosines. Questions will be evenly split between SAS and SSS. • Circulate as students are working to answer questions and ensure all students are staying on task.

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 during and after the lesson to ask if any students have questions about the material. • As an additional formative assessment, students will solve an additional nine oblique triangles, again using the Law of Cosines. Five questions will involve SAS, while four will involve SSS.

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 nine problems).

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?

• Due to time constraints as well as the struggles of virtually all students with the algebra necessary to solve an oblique triangle given all three side lengths (SSS), students were instructed to complete only the first five questions on the classwork (though they were asked to complete the homework in its entirety). All but two students got stuck on the algebraic manipulations necessary to rearrange the standard form of the Law of Cosines (SAS) to solve for a missing angle measure, though once students were guided through this derivation, all were able to successfully solve example SSS problems. Regardless, I will include additional practice of this at the start of class tomorrow. • Among the first five problems on the classwork, only three students mixed up the angle measures due to solving in the incorrect order (not solving for the largest angle first); the remaining students all correctly answered all five problems. This suggests, as expected, that students are generally comfortable with the SAS case of the Law of Cosines; this is also a somewhat surprising (and refreshing) result, since this is typically the more complicated of the two cases (SAS and SSS).

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 happened in the previous lesson, the Geometer’s Sketchpad derivation of the Law of Cosines took more time than I had planned; in addition, after consulting with my mentor, I realized it had been too complex and abstract for the students. While I had reviewed the demonstration the previous night and did not expect it to take too much time, in retrospect, I should have noticed that it was going to be too complicated for students—or spoken to my mentor about it beforehand—and come up with a less complex derivation. I do feel that it is important for students to be exposed to these derivations (even in a level six class), but I also realize there are times when students are being asked to make too much of a leap based on their existing knowledge base and ability levels. This is the last Sketchpad demonstration I am planning to use in this unit, but I am planning on more demonstrations in subsequent units; I will be sure to carefully review these to make sure I am not asking students to reason at too abstract a level. • One potential complication with the homework that I had not foreseen is that due to the structure of the assignment (matching answers to letters to solve a trivia question), students do not need to fully solve each triangle, but can instead find only one of the missing side or angle measures and then fill in the blank. One way to avoid this would have been to have answers that differ only in the measurement of their final unknown; given the enormous complexity of this, however, it would likely not be practical. A more realistic approach would have been to split the assignment in half: have the students solve each triangle for homework tonight and then match up their answers to the trivia letters as a warm-up tomorrow. I do enjoy utilizing assignments of this type rather than simply straight calculations, but I do now realize that some accommodations are necessary for problems that require multiple calculations, such as those featured here. (If this was an algebra class, for example, and


there was only one value rather than three for each question, this would likely not be as much of a concern.)

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 base the instruction on the pretest, previous formative assessment, and students’ prior knowledge; in addition, I make sure to take into account what worked and what didn’t in the previous lesson. Solving oblique triangles is a relatively obscure topic, and students do not yet have the full knowledge foundation necessary to solve application problems; nonetheless, I made sure to begin with an example to which students could legitimately relate (laying down a bunt)—one that was particularly applicable to the baseball and softball players in the class. This lesson also included considerations for a student who missed her fourth consecutive day of school today; while I have been fortunate enough to have otherwise perfect attendance thus far (though it is only day two of five), 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 this student up on the material she has missed. In addition, as was stated earlier, the derivation of the Law of Cosines using Geometer’s Sketchpad required extremely rigorous calculations—in fact, as my mentor and I determined following the lesson, it ended up being too rigorous for the students in this academic-level class. Though not an ideal outcome, this does show the rigor and diversity considerations that went into this lesson. • InTASC 8 (Instructional Strategies): This lesson reflects the influence of contextual factors in planning a lesson: Although part of the lecture portion of the lesson is devoted to calculator skills such as finding the angle corresponding to a given cosine value, 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 of their own on which to practice these strategies. I also intentionally used an example at the start to which all students can relate: regardless of socioeconomic status, all students can be expected to be familiar with baseball and softball since WMHS sponsors varsity teams in both sports. The biggest reason for this was to ensure that students would find the content accessible and able to relate to it: They likely have never needed to find the missing side in a random triangle in their everyday lives, but everyone who has played a game of baseball or a round of golf, or needed to find a ship’s distance from land (an activity with which our JROTC students are familiar), can use the strategies introduced in today’s lesson. My goal is always to make content as meaningful and accessible as possible for students, and I feel this lesson went a long way towards doing just that.


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 February 26, 2014 Mentor Mark Heffner Standard(s): ____ CCSS

Unit Solving Oblique Triangles Grade Class Size 11/12 16 School Winters Mill High School

Topic Law of Cosines: SSS Time 7:30-8:50 a.m. Intern Ben Cohen

Other: Maryland Content Standards

2.2.2.1 The student will apply the Law of Sines and the Law of Cosines to solve problems involving oblique triangles. Big Idea or Essential Question Today’s lesson continues the focus from yesterday on the second method for solving oblique triangles, the Law of Cosines. Today’s lesson will focus on the second case in which the Law of Cosines is used: all three side lengths, but no angle measures, are known (SSS). This is the only one of the five potential cases that does not involve both side and angle measures and is thus the most complex of the five. In addition, part of this lesson will be devoted to practice of solving triangles given all five of these scenarios. The gradual process of exposure to the outside connections of these two laws (Sines and Cosines) will also continue in this lesson; students will begin tackling these problems in the following lesson, so they will continue to be exposed to them in an attempt to set them up for success in future lessons.

Alignment with Summative Assessment Questions 10 and 12 (SSS) on the CCPS summative assessment ask students to solve an oblique triangle using the Law of Cosines given the lengths of all three sides. Question 10 includes a sketch of the triangle, while question 12 includes only the side lengths. Lesson Objective Students will use the Law of Cosines to solve an oblique triangle in which all three side lengths but no angle measures are known. Formative Assessment Students will be asked to complete a series of problems for classwork and homework; for today’s class, however, there will be a single handout that students will begin in class and finish for homework. This handout will include 18 problems for students to complete; they will be split evenly between the Law of Sines and the Law of Cosines. In addition, these problems will be evenly split between ASA, SAA, SSA, SAS, and SSS scenarios. Students will also complete the final four problems from the previous night’s homework. Informal assessments are integrated throughout the lesson; remediation of past material will be included as the new material is introduced if it becomes necessary.


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.

• No students correctly answered question 4 on the pretest, which was a free-response question asking them to solve an oblique triangle using the Law of Cosines given all three side lengths.

• This shows students have no prior knowledge of the Law of Cosines and are unfamiliar with how to use it to solve triangles given SSS. As expected, I will have to introduce this as entirely new material.

• Only three out of 16 students incorrectly solved problems on yesterday’s classwork due to finding the angles in the incorrect order; the remaining incorrect answers were due to simple arithmetic errors.

• This is the most frequent error when solving triangles using the Law of Sines; it is gratifying to see that the vast majority of students have made it past this sticking point. More importantly, this suggests students have mastered an earlier concept with which many initially had trouble: knowing that cosine is negative in Quadrant II by definition.

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

• Many students encountered significant difficulty attempting to manipulate the equations for the Law of Cosines for the situation in which the angle is unknown (side-side-side).

• This is the first lesson in the unit to not feature a Geometer’s Sketchpad demonstration, given that yesterday’s demonstration proved to be too advanced and abstract for the students. (This decision was made in consultation with my mentor following yesterday’s lesson.)

Individual or Small Group Needs Specific to this Lesson

• Students were instructed to complete only the first five problems on the classwork, since these dealt with side-angle-side scenarios. In addition, I will rederive the equations for finding an unknown angle at the beginning of class today, devoting more time to this than was allotted during the preceding lesson. • As I essentially ended up scrapping yesterday’s demonstration and running through it in a more concrete manner, I do not expect this to have any negative consequences for students’ learning. At the same time, I will have my computer and Sketchpad ready in case it appears students need to see this topic approached in a different light.

►►►►►

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

• L. has missed the first two days of this unit due to illness. (These are the first class days she has missed all semester.)

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

• I will need to work with her one-on-one during either FLEX time or after school as time and scheduling permit. As she is one of the strongest students in the class, I expect she will have minimal difficulties catching up, but this should nonetheless be done sooner rather than later.


• Care must be taken to engage all learners, being mindful of the gender split and high percentage of FARMS students.

• As is done in all lessons, any real-world examples will be as universally applicable as possible and will avoid making reference to luxuries that some students may not have experienced. Also, any discussions of calculator techniques must be carefully framed—rather than asking which students own a TI-83 or TI-84, for example, I will simply preface this with “If you have a TI-83 or a TI84…”

III. Instructional Procedures – What instructional strategies will I use to ensure that every child is a successful learner? Instructional Materials/Resources The instructor will prepare a handout with 18 problems on it to be started as classwork and finished as homework. All problems will ask students to solve a given oblique triangle; students will need to use both the Law of Sines and the Law of Cosines to solve the triangle. This lesson will also make extensive use of the TI-Nspire calculators located in the classroom.

Technology Integration When solving oblique triangles, extensive use will be made of the TI-Nspire calculators in the classroom. This is particularly necessary for this class, which in many cases will require students to go backwards and find the angle corresponding to a pair of given side lengths. The lesson will also include a component describing for students how to perform these calculations on a TI-83 or TI-84. Cross-curricular Connections Students will be exposed to a variety of applications to other fields that require the use of the Law of Cosines. As was done in previous lessons, examples will be drawn from various fields, including architecture (finding the height of a building or broadcast antenna), meteorology (finding the distance from the U.S. coastline of a storm at sea), maritime navigation (finding the distance between two ships and land), and sports (determining the best way to play a golf hole). Students will be exposed to these applications, but will not be expected to solve such problems until Thursday’s lesson, which will introduce them to the techniques for solving such problems. 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 equations 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. • The administration has hinted that there may be an emergency drill this week and asked all teachers to remind students of these procedures. As such, it is possible class may be interrupted for an extended period of time one day this week. • 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 5 minutes

Warm-up Motivation Bridge

Procedure • For the warm-up, students will be asked to use the Law of Cosines to solve an oblique triangle. This will be a side-angle-side scenario; in addition, students will be given the proper order to use when finding the unknown terms to avoid problems that befell them in the previous lesson. • Today’s lesson will focus on the second application of the Law of Cosines: triangles in which all three side lengths are known, but no angle measures are given. In addition, students will be given time to practice solving triangles using both the Law of Sines and the Law of Cosines.

Development of the New Learning

45 minutes

(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.)

• In the previous lesson, we derived the equations for using the Law of Cosines when one side is unknown (i.e., side-angle-side). Today’s class will begin with a derivation of the equation for finding an unknown angle using the Law of Cosines; this is simply algebraic manipulation, but many students encountered significant difficulties with it. • Start with the equation for the Law of Cosines when side a is unknown. Now, suppose it is angle A that is unknown. We can manipulate the equation to isolate the term with angle A in it; this gives us

cos A =

a2 − b2 − c2 . We can then use the inverse −2bc

cosine function to find the angle measure corresponding to angle A, and use a similar derivation to find the equations for when angle B or angle C is not known. • Inform students that there is one pitfall they must be extremely careful to avoid when solving for an unknown angle. In order to tell whether a triangle is acute or obtuse, we must find the smallest angle first when using the Law of Cosines. (We know that the smallest angle will be opposite the shortest side.) Similarly, when we then use the Law of Sines, we would need to use the secondshortest side and second-smallest angle. This will ensure we know if the largest angle is acute or obtuse. • Go through two examples in which students are given three side lengths but no angle measures. The first will be an acute triangle, while in the second, the largest angle will be obtuse. Enrichment or Remediation (As appropriate to lesson)

25 minutes

• Ensure at this point that students are familiar with the proper order to follow when it is necessary to begin solving an oblique triangle with the Law of Cosines: this method is used first, followed by the Law of Sines to find one unknown angle; the final unknown angle can be found by subtracting the sum of the two known angles from 180 degrees. • Remind them also of the situations in which each law is used. If


we are given SSA, ASA, or SAA, only the Law of Sines is necessary; for SAS and SSS, we must begin with the Law of Cosines. • For classwork, students will complete 18 problems asking them to solve an oblique triangle using the Laws of Sines and Cosines. Questions will be evenly split between the two laws as well as between SSA, SAA, ASA, SAS, and SSS. Students will complete this assignment for homework as well. • Circulate as students are working to answer questions and ensure all students are staying on task. 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 during and after the lesson to ask if any students have questions about the material. • As an additional formative assessment, students will solve an additional four oblique triangles, again using the Law of Cosines. All of these will be SSS triangles left over from the previous lesson’s homework, on which students were asked to complete only the first five problems.

Planned Ending (Closure) • •

5 minutes

Summary Homework

• 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. • Allow students time to begin the homework at the end of class.

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?

• On the formative assessment, five students made the mistake of applying the Law of Sines to find the remaining two angles (after using the Law of Cosines) in the wrong order, solving for the larger remaining angle first rather than the smaller remaining angle. It is possible that they mixed this up with the procedure for the Law of Cosines in the SAS case, when they must solve for the larger remaining angle first. Five out of 16 students is actually a relatively small percentage given the potential frequency of this error—it is a very easy mix-up to make—so while I will certainly have to continue to be on the lookout for this going forward, it does not appear it will be as much of a concern as I had originally feared.


• Through the first three lessons of this unit, student achievement has been exceptional—even higher than I had anticipated. I have felt that this is one of the three most difficult units students will tackle in this class (along with periodic functions and polar coordinates), but they have certainly impressed me with their quick acquisition of the material. This is particularly important since it allows me to spend the final two lessons on applications and other interactive demonstrations and activities without having to worry about remediation (while new material will be introduced in these lessons, area formulas—the topic of the final two classes—are very simply, and given their achievement thus far, students should have no problems grasping them). 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?

• This lesson in particular reminded me of the importance of willing to be flexible and adjust “on the fly.” Although I was originally planning to begin area formulas today, I also had to push back the SSS case of the Law of Cosines due to the time yesterday’s lesson took; my mentor and I decided it would be best to devote the rest of today’s class to mixed practice rather than introducing one area formula and potentially having to split that between two classes as well. While we have no reason to suspect the students are in need of remediation, extra practice is always beneficial, and this allows me to use today’s lesson to get back on track rather than setting up another situation in which I have to split a topic across multiple lessons. • It was helpful for both the students and me to guide them through the algebraic derivations necessary to rearrange the Law of Cosines to find a missing angle, rather than expecting the students to be able to do it on their own. As I have repeatedly stated, while there are some skills that students in Trigonometry/Pre-Calculus are expected to have mastered, that does not mean these expectations will necessarily be the reality; in a level six class, it is perfectly acceptable for students to ask for help in performing these derivations. It was also helpful for me, as it allowed me to continue moving through the unit at a reasonable pace rather than getting bogged down trying to find the source of students’ struggles with the material. This was particularly acute since the class was slightly behind the pace I had originally hoped for at the start of the unit. • Although students will not learn the techniques for solving application problems until the following two lessons, exposing them to these applications is always helpful. This ensures continued student interest and engagement and avoids the trap of teaching them the material in a vacuum, which is a particular concern in a lesson or series of lessons relatively heavy on foundational material such as these.


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 9 (Professional Learning and Ethical Practice): Much of the activities and structure of this lesson were designed to allow me to gauge my instruction at the midway point of the unit. 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). By devoting this lesson to mixed practice, I could get a handle on how successful students had been at retaining the knowledge they had demonstrated in previous lessons; in turn, this allowed me to judge whether it would be necessary to include any remediation of concepts in subsequent lessons or whether students had acquired sufficient knowledge to move forward within the unit. I typically do such a lesson halfway through every unit to ensure I am keeping my finger on the pulse of how students are faring, though normally this is done through an activity within the lesson (such as a strategically designed series of warm-up questions) rather than being a partial focus of the class. 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.


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 February 27, 2014 Mentor Mark Heffner Standard(s): ____ CCSS

Unit Solving Oblique Triangles Grade Class Size 11/12 16 School Winters Mill High School

Topic Area of Oblique Triangles Time 7:30-8:50 a.m. Intern Ben Cohen

Other: Maryland Content Standards

2.2.2.1 The student will apply the Law of Sines and the Law of Cosines to solve problems involving oblique triangles. Big Idea or Essential Question This will be a two-part lesson designed to instill in students a greater appreciation for the underpinnings of the mathematics they have studied in this course thus far. In the first part of the lesson, students will be divided into groups, with each receiving a different textbook excerpt on the history of some mathematical topic related to this course (e.g., the use of trigonometric functions by Copernicus and Kepler in deriving their heliocentric models of the solar system). After reading, students will fill out a graphic organizer and then share their thoughts with the class. This will conclude with a look at the biography of Heron of Alexandria, for whom one of the area formulas for oblique triangles is named; these two area formulas will be the focus of the second part of today’s lesson.

Alignment with Summative Assessment Questions 2, 4, and 14 on the CCPS summative assessment ask students to find the area of an oblique triangle. Question 2 gives two sides and the included angle, while question 14 gives three side lengths; question 4 gives two angles and the non-included side, requiring the use of the Law of Sines in addition to the area formula. Lesson Objective Given two sides and the included angle or three sides of an oblique triangle, students will find the area of the triangle. Formative Assessment Students will be asked to complete a handout containing 10 problems for homework. All problems will ask students to use one of two formulas to find the area of a given oblique triangle, with five problems devoted to each formula. In addition, today’s content-literacy activity will serve as a formative assessment; although it does not directly assess students’ knowledge of the material in this unit, this knowledge is essential to draw the cross-curricular connections that are the focus of the lesson. Informal assessments are integrated throughout the lesson; remediation of past material will be included as the new material is introduced if it becomes necessary.


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.

• Six students correctly answered question 5 (multiple choice) on the pretest, which asked them to find the area of an oblique triangle; none, however, correctly answered free-response questions 6 or 7 on the same topic.

• The bulk of students guessed “C” for all multiplechoice questions; as this was the correct answer to question 5, it is likely that these correct answers are the result of random guessing, and questions 6 and 7 give a better picture of students’ abilities. As such, it appears students have no prior knowledge of the area formulas, as expected, and so it must be introduced as entirely new material.

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

• The content-literacy portion of today’s lesson will be entirely new to students. In addition, some readings include advanced mathematics well beyond what would be expected of a high-school trigonometry class.

Individual or Small Group Needs Specific to this Lesson

• Students have been informed in advance of the different format the first part of today’s lesson will take. In addition, I will warn them tomorrow that some of the mathematics in the text will be far beyond what they can be reasonably expected to know and instruct them to consult with me or my mentor teacher if they have questions.

►►►►►

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

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

• M. returned to class on Tuesday for the first time in nearly two weeks (illness). She has a substantial amount of work to make up.

• Despite this, she has seamlessly adjusted to this unit and received a perfect score on Tuesday’s formative assessment. Although I must still remain aware of the possibility that she will miss another extended block of time due to illness, it appears my concerns about extended remediation being needed may have been unfounded.

• Care must be taken to engage all learners, being mindful of the gender split and high percentage of FARMS students.

• As is done in all lessons, any real-world examples will be as universally applicable as possible and will avoid making reference to luxuries that some students may not have experienced. Also, any discussions of calculator techniques must be carefully framed—rather than asking which students own a TI-83 or TI-84, for example, I will simply preface this with “If you have a TI-83 or a TI84…”


III. Instructional Procedures – What instructional strategies will I use to ensure that every child is a successful learner? Instructional Materials/Resources The instructor will prepare a handout with 10 problems on it to be started as classwork and finished as homework. All problems will ask students to find the area of a given oblique triangle; there will be five problems for each formula. In addition, the instructor will prepare a graphic organizer and four different handouts (passages from History of Mathematics by Victor Katz) for use during the content-literacy portion of the lesson. This lesson will also make extensive use of the TI-Nspire calculators located in the classroom.

Technology Integration When finding the area of oblique triangles, extensive use will be made of the TI-Nspire calculators in the classroom. This is particularly necessary for this class, like the two before it, which in many cases will require students to go backwards and find the angle corresponding to a pair of given side lengths (the angle formulas discussed today make extensive use of the Law of Cosines). The lesson will also include a component describing for students how to perform these calculations on a TI-83 or TI-84. Cross-curricular Connections The first part of today’s lesson will focus primarily on the history of mathematics and the connections that have been drawn between the material being studied in this class throughout history—for example, using the Laws of Sines and Cosines as evidence for a heliocentric model of the solar system. One reading will also focus on contemporary uses of technology in mathematics and the advancements made possible by this. These historical connections will be integrated in future units whenever circumstances permit. Management Considerations (Procedures, Transitions, Materials) and Student Engagement (required) • This will be the first time this semester that students have worked together in groups on a class assignment. Both my mentor and I will need to be vigilant during this portion of the class to ensure that students are staying on task and that all group members are contributing. In addition, this will be the first class discussion in some time (following the jigsaw strategy); I will again need to ensure that all students are participating and have questions ready to jump-start the discussion if it begins to stall. • The administration has hinted that there may be an emergency drill this week and asked all teachers to remind students of these procedures. As such, it is possible class may be interrupted for an extended period of time one day this week. • 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 • • •

Warm-up Motivation Bridge

Approximate Time 5 minutes

Procedure • For the warm-up, students will be asked to use the Law of Cosines to solve an oblique triangle. This time, students will be given all three sides of the triangle (SSS).


• Today, students will learn how to use a pair of formulas to find the area of an oblique triangle. This will be the second part of class, however; the first part will be devoted to the integration of a content-literacy component introducing students to topics in the history of mathematics that have had a pronounced impact on the material they are currently studying. Development of the New Learning

30 minutes

(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.)

(NOTE CHANGE OF SEQUENCE) • Students are aware of how to find the angle of a right triangle: the familiar formula

1 A = bh . This does not apply to oblique 2

triangles, however. How can we find the area of these triangles? • Start with an example of a triangle for which two sides and the included angle are given (SAS). Using our knowledge of trigonometric identities and substituting into the above formula, we find that

1 A = bcsin(A) . Similarly, we can also use sides a and 2

b and angle C or sides a and c and angle B, depending on which sides and angle are initially given. • Take students through two examples of using this area formula. While it is possible to give two sides and the non-included angle and use the Law of Sines to find the included angle, in all examples, students will be given the included angle to ensure the emphasis is on finding the area. • What about if we are given three side lengths? Certainly, it is possible to use the Law of Cosines to find the included angle and then use the area calculation outlined above. A far easier method,

A = s(s − a)(s − b)(s − c) , 1 where s is the semiperimeter, defined as s = (a + b + c) . Once 2 however, is to use Heron’s formula:

again, take students through two examples of this. • Allow students time at the end of class to begin working on the homework: 10 problems asking students to find the area of an oblique triangle. Five problems will use the first area formula, while five will require the use of Heron’s formula. Enrichment or Remediation (As appropriate to lesson)

30 minutes

(NOTE CHANGE OF SEQUENCE) • Students will begin today’s class with a content-literacy activity designed to provide them with an understanding of the historical foundations of the mathematics they are studying in this class. For this, students will be given a series of excerpts from History of Mathematics (Victor J. Katz). Students will be broken into four groups of four each; selections will focus on the development of Euclidean geometry (including the Pythagorean theorem), Renaissance mathematics (including the trigonometric derivations used by Copernicus and Kepler for their heliocentric models of the solar system), and contemporary technology used in mathematics.


• Students will initially read their assigned selections in their assigned groups and will fill out a graphic organizer as they read asking them to summarize the big ideas or major developments, their effects at the time and today, and their relevance to the material we have studied thus far. When all students are finished reading, students will be asked to regroup according to their assigned readings and exchange ideas and thoughts. They will then return to their original groups and share their findings with their fellow group members. • After students have had sufficient time to share their thoughts, the instructor will survey them to get a sample of thoughts on each reading. These will be added to a full-class graphic organizer. • Circulate as students are working in groups to answer questions and ensure they are remaining on task. 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 and also in their reading groups, the instructor will circulate to answer questions and ensure all students comprehend the material and are remaining on topic. • Take time during and after the lesson to ask if any students have questions about the material.

Planned Ending (Closure) • •

5 minutes

Summary Homework

• 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. • Allow students time to begin the homework at the end of class.

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?

• As expected, students almost instantly grasped the area formulas. A few students did struggle somewhat with SSA problems (which require an additional step before the sine-based area formula can be used), though 100 percent of students were otherwise able to solve area-based problems. • With respect to the mixed practice, two students (H. and E.) correctly answered every problem; the most frequent source of difficulty for the remaining 14 students was SAS (ASA, SSA, SAA, and SSS were generally fine). While SSS is generally thought of as the most complex of the five possible scenarios, SAS can also be quite difficult, so this is not a surprising result. It is good to see that students have both generally mastered the material in this unit and have maintained that content knowledge; this also validates


the devotion of a day to mixed practice rather than introducing a new topic that would again straddle two classes. • Only one student (L.) out of 16 fully filled in their graphic organizer for the content literacy lesson. This will be discussed in further detail in the following section, but while it is clear that the students did put serious effort into the activity, I realize the time they were allotted was insufficient, and that I should have planned this activity to take the entire class rather than simply half of it. 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 stated earlier, I failed to devote sufficient time to the content literacy lesson. While my mentor had previously used a similar strategy with this unit in the past, this was a more intense and involved activity than what he had done, so I should have reevaluated the amount of time devoted to it instead of simply going with what he had done in the past. The idea behind the activity was to expose students to the history and applications of the material they are currently studying, as well as help them tie together all of the material from their studies over the prior four years, but it appears the readings I chose were at a level of mathematics too advanced for them. (While the readings were all taken from a college-level history of mathematics textbook, I attempted to select passages that concerned material with which students were already familiar, such as the Pythagorean theorem, as well as contemporary issues such as computers and calculators.) In addition, some resistance from students was to be expected since this is an entirely new activity for them—while they are presumably used to doing similar activities in English and possibly social studies classes, they have almost certainly never done anything like this in their previous math classes. I continue to strongly feel that reading and writing activities in math class are extremely worthwhile, and students’ responses both verbally and on their graphic organizers suggest that they were able to derive significant utility from this lesson, but I will need to tweak the structure of this activity before implementing it with future classes.

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 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 readings used in this lesson were carefully selected to ensure they represented topics to which all students could relate, regardless of socioeconomic status. (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 that draw on their past schooling and


other items to which there is universal exposure, rather than including some topics that may have been unfamiliar to low-SES students, I hoped to instill in the full class an appreciation for the history and power of the seemingly utility-free topic of oblique triangles. • InTASC 5 (Application of Content): This lesson includes an introduction to the myriad applications of oblique triangles, with applications in fields as disparate as astronomy, technology, and psychology. Many of these, such as the Turing test, represent situations to which students can legitimately relate—for example, every student in the class has witnessed the power of modern technology such as Geometer’s Sketchpad, which I have used regularly throughout the semester. Opportunities for students to brainstorm their own applications of oblique triangles 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 this seemingly irrelevant topic has on their everyday lives.


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 February 28, 2014 Mentor Mark Heffner Standard(s): ____ CCSS

Unit Solving Oblique Triangles Grade Class Size 11/12 16 School Winters Mill High School

Topic Applications of Oblique Triangles Time 7:30-8:50 a.m. Intern Ben Cohen

Other: Maryland Content Standards

2.2.2.1 The student will apply the Law of Sines and the Law of Cosines to solve problems involving oblique triangles. Big Idea or Essential Question This will be the second of two lessons designed to instill in students a greater appreciation for the applications of the mathematical techniques introduced in this unit. Today’s lesson will also include a significant interactive component: Students will be given the opportunity to measure triangles formed by objects inside or outside of the school building and then find the area of these triangles, a miniature version of calculations that took place involving these same techniques thousands of years ago. Students will also be exposed to a contemporary real-world application of this: the Bermuda Triangle. A brief reading will be distributed to students, and they will also be asked to think about additional applications of these mathematical techniques.

Alignment with Summative Assessment Questions 2, 3, and 13 on the CCPS summative assessment all ask students to solve problems involving applications of oblique triangles and the techniques covered earlier (Law of Sines, Law of Cosines, and area formulas). Lesson Objective Students will use Heron’s formula and the sine formula to solve problems involving applications of oblique triangles, including finding the areas of triangles formed by objects in and around the school building. Formative Assessment The activity that will form a focal point of the lesson is also designed to serve as a formative assessment—rather than directly giving students triangles and asking them to find the area of each, I will see how effective they are at both measuring triangles and finding their respective areas. (Although I do not have the exact values for the area of each, I can measure the accuracy of each group with respect to the others.) Informal assessments are also integrated throughout the lesson; remediation of past material will be included as the new material is introduced if it becomes necessary. Lastly, this lesson will include the summative assessment for the unit as well.


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.

• All students were successfully able to find the area of oblique triangles using both the sine formula and Heron’s formula; while three students did not complete the worksheet in its entirety, all of their answers were correct.

• In addition to showing mastery of the content, this suggests that students are well prepared for today’s activity of finding the area of triangles in and around the school building. They should thus have no problems with this activity; no remediation should be necessary.

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

• There were no problems with off-task behavior during yesterday’s group activity, which was the first time this semester students were placed into groups for a class activity.

• Some students did not take the post-test seriously during my fall ESL (since it did not count for a grade), as evidenced by both students’ comments and the divergence between their achievement on the formative assessments and the post-test.

Individual or Small Group Needs Specific to this Lesson

• While today’s logistics will be slightly more challenging since students will be spread out (though still in the same general area), it appears I do not need to worry about anyone wandering off or otherwise not doing what they are supposed to. This would certainly be a concern for a foundational class, but it should not be a problem for these students. • I will again not count this post-test for an actual grade and will again count on the students to put in a solid effort. While this may appear to be a flawed strategy, it is clear that a majority of students put in full effort on yesterday’s content literacy lesson despite their readily apparent disdain for the activity; I expect a similar showing from them today. In some instances, there is nothing that can be done about this; I will, however, note cases in which a student’s post-test score deviated sharply from his/her formative assessment achievement.

►►►►►

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

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

• Some students (particularly E. and L.) are perfectionists and may become concerned if their measurements are off in the slightest from those of another group.

• I will emphasize at the beginning of the lesson that these are approximate, not exact measurements, and it is not at all expected that every group’s measurement will be the same. In addition, there is no single correct answer since each group will likely use a slightly different starting and ending point.

• Care must be taken to engage all learners, being

• As is done in all lessons, all examples used today


mindful of the gender split and high percentage of FARMS students.

will be as universally applicable as possible and will avoid making reference to luxuries that some students may not have experienced. Also, any discussions of calculator techniques must be carefully framed—rather than asking which students own a TI-83 or TI-84, for example, I will simply preface this with “If you have a TI-83 or a TI84…”

III. Instructional Procedures – What instructional strategies will I use to ensure that every child is a successful learner? Instructional Materials/Resources I have borrowed multiple long-distance tape measures from other teachers in the department to ensure students will be able to efficiently measure these relatively long distances. If it becomes necessary to move the activity indoors due to weather, I will mark locations using tape and colored stickers. This lesson will also make extensive use of the TI-Nspire calculators located in the classroom. In addition, the post-test for this unit will be given at the end of the lesson. I have also prepared a brief handout on the Bermuda Triangle to give to students at the start of class; while we will not discuss this to the same extent as yesterday’s readings, given today’s time constraints, I do feel it is important for students to be exposed to another application of this material.

Technology Integration When finding the area of oblique triangles, extensive use will be made of the TI-Nspire calculators in the classroom. The lesson will also include a component describing for students how to perform these calculations on a TI-83 or TI-84. There is also a significant low-tech component to this lesson: students will need to use a long-distance tape measure to find the lengths of every triangle; this will also help assess their ability to measure these distances and attend to precision. Cross-curricular Connections The measuring activity during the first part of today’s lesson is a smaller-scale version of how oblique triangles were used in ancient times—although it would not be feasible to have students determine the distance from Westminster to Frederick and the Pennsylvania state line and then calculate the area of the triangle formed by the proverbial lines, I will be sure to mention that this method was used in Ancient Greece to evaluate the acreage of a given parcel of land. The Bermuda Triangle article that will be distributed at the start of class is another connection, and I will ask students to brainstorm additional potential applications following the activity. Management Considerations (Procedures, Transitions, Materials) and Student Engagement (required) • Students worked in groups yesterday, to great success; there were no issues with off-task behavior. However, today will present a bigger management challenge: Instead of being in the classroom, students will be spread out (though still in the same general area) throughout the school grounds. Again, both my mentor and I must be vigilant and ensure all students are staying on task rather than wandering off or holding conversations with friends in the hallway. Although I do not foresee this being a problem with this class (academic level consisting of juniors and seniors), I cannot simply assume students will behave and leave them to their own devices; I need to be proactive to make sure this is the case. • The current plan is to have students measure distances between landmarks outside the school (such as the flagpole, stadium/track entrance, and sign). However, the weather forecast is calling for a wind chill in the single digits during the early morning, when this class meets. If this is the case, I will not be able to bring the class outside; as a backup plan, I will mark off three triangles inside the building prior to the start of class and will have students measure these triangles instead if it is too cold to bring them outside.


• The administration has hinted that there may be an emergency drill this week and asked all teachers to remind students of these procedures. As such, it is possible class may be interrupted for an extended period of time one day this week. • 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 performing these calculations 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 the warm-up, students will be asked to find the area of a given oblique triangle using Heron’s formula. Although there are challenging variations of this, in this instance they will simply be given the three side lengths. • At the start of class, pass out a copy of an article on the Bermuda Triangle for students to read. This will be followed by a discussion on how to use the techniques we have covered thus far to answer some of the key questions that are posed in the reading (less formal than yesterday’s discussion), such as why, with today’s advanced search technology, so many planes and ships can simply disappear without a trace (use Heron’s formula to find the area of the Bermuda Triangle: over 1.5 million square miles). • Today’s lesson will be an interactive one, allowing students to measure the distances between a series of points outside the WMHS school building and calculate the areas of the resulting triangles. This will continue the study of applications of oblique triangles that we began in yesterday’s lesson.

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

• Break students into four groups of four. For today’s lesson, they will be asked to measure the distances between a series of “landmarks” outside of the school building and use these measurements to find the area of the resulting oblique triangle. The “landmarks” are as follows: • Electronic sign, flagpole, and entrance to student parking lot • Entrance to stadium/track, entrance to faculty parking lot, and light pole in faculty parking lot • Main school entrance, front bench, and flagpole Note: If it is too cold to bring students outside, they will measure three alternate triangles inside marked with the use of taped “X”s and colored dots. These triangles will be placed in the main hallway, rotunda, and cafeteria. • Circulate as groups are working to ensure all students are remaining on task and answer any questions that may arise. In addition, keep an eye on the progress of each group.


• It is anticipated that each group’s measurements will differ slightly due to differences in starting and ending points; this is normal, but some students may question whether their measurements are accurate if they are slightly off compared to those of another group. • When all groups have finished, return to the classroom so groups can share and analyze their findings and debrief. Enrichment or Remediation

15 minutes

(As appropriate to lesson)

• Have all groups share their measurements for the three triangles in order. Survey students as to the possible causes of the expected slight variation between each group’s measurements (less precision, differences in starting and ending points, and human error, to name a few). • After this, explain to students that this method was used quite frequently in past societies (such as Ancient Greece) to survey parcels of land—since multiple triangles can be put together to form a square, this proved to be a useful method of measurement. • Survey students to see if they can come up with some other fields in which oblique triangles may play a significant role. (Some examples include the construction of the Pyramid Arena in Memphis and basketball’s triangle offense.)

Assessment/ Evaluation

Ongoing

• Students will be informally assessed throughout the lesson. During the measurement activity, the instructor will circulate to answer questions and ensure all students are remaining on topic. The discussion portions of the class will also serve as an informal assessment; they will be used to evaluate the extent to which students can reason abstractly and extend their knowledge to solve application problems. • 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. • The final 20 minutes of the period will be devoted to the post-test for this unit. Students will be permitted to use a calculator on this assessment.

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

• Since there were no formal assessments aside from the posttest, there is not a large amount of data from which to draw. However, as was stated earlier, I used the measuring activities as a formative assessment; based on this, all four groups successfully measured the triangle in the cafeteria, and three out of four successfully measured the triangle in the rotunda and main hallway (there were two different groups with erroneous


need for the class as a whole, subgroups or individuals? What will be your next steps based on this analysis?

measurements). As these groups applied the correct area formula, it appears they erred in their measurement of the sides of each triangle. (As expected, there was some slight variance in each group’s measurement due to different starting and ending points). • In all, it appears this unit is an area of strength for the class. Based on their success on the previous unit, solving right triangles (14 out of 16 students scored a 94 percent or higher on the summative assessment), this is not entirely unexpected; however, it should also be acknowledged that this is a significantly more complex unit, even though it relies on a similar skill set. This observation will be confirmed by an analysis of the post-test data.

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 expected, it was too cold to bring students outside due to a morning wind chill of 8 degrees. Instead, I had them measure triangles in the rotunda, main hallway, and cafeteria (all of which are observable from the main entrance). While there were no “landmarks” to use, I did try to strategically place the triangles whenever possible—such as making one have its vertices at the water fountain, trophy case, and cafeteria entrance. This was an enormously successful activity; in addition to exposing students to additional applications of the material, I was able to get everyone engaged and allow them to acquire additional knowledge in a setting other than the classroom. I always attempt to ensure class is not becoming too repetitive or predictable, and this was a very useful way to continue these efforts. • Surprisingly to me (and my mentor), virtually no students in the class were familiar with the metric system as it pertains to length measurement (they did have some familiarity with volume measurement, having used liters in chemistry class). Though not originally planned, I added a roughly 10-minute introduction to, and discussion of, the metric system before the post-test; in addition to explaining to students how it is based on increments of 10, like our number system, we discussed why the U.S. is one of the few remaining countries still using the imperial system of measurement. Again, this was a very useful activity as far as getting students to think beyond merely solving problems and push themselves to develop a deeper understanding of the course content, even though the metric system and its adoption is not directly related to 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 3 (Learning Environments): I go to great lengths to ensure that students are actively engaged in all lessons, but this took it to a new level. This is the first lesson I have done in over six months at WMHS that allowed students to participate in a hands-on activity outside of the classroom—not only do these take considerable time to plan and execute, but I also feel that they should be used sparingly, as a treat, so that students do not become accustomed to them. In addition, there was a significant collaborative learning component to this activity: students were asked to work in groups to measure the


respective distances and come up with the area of the resultant triangle. The emphasis on applications of the area formulas and other oblique-triangle calculations also must be taken into account here: Students were asked to both consider how oblique-triangle trigonometry can help solve some of the questions surrounding the Bermuda Triangle as well as other fields in which the techniques in today’s lesson (or any of the four previous ones) could be helpful. Above all, this was the second consecutive lesson with a significant discussion component; this is another key tenet of my goal to create a learning environment in which all students feel comfortable speaking openly without fear of making a mistake or being ridiculed. • InTASC 4 (Content Knowledge): Unquestionably, the material in this unit is fairly dry. This is particularly true for the two area formulas (sine and Heron’s formula), since unlike their parent laws, they do not readily relate to students’ past work in geometry. For this reason, I felt it was extremely important to expose students to not only the history of these formulas (as was done in yesterday’s class)—which goes back thousands of years—but also some of their applications. Although students lack the mathematical background to work out any such application problems—which would require years of advanced study—it was crucial for them to learn that there do exist myriad applications of these formulas. In addition, I have attempted throughout this unit to link new ideas to familiar ones, and this certainly fulfills that aim. As I have remarked to students multiple times, this unit is one of the three most difficult in the Trigonometry/Pre-Calculus curriculum; by relating the content to concepts with which students are innately familiar, such as the Bermuda Triangle, I hope to make it easier for students to understand both the content and how it affects their everyday lives. • InTASC 10 (Leadership and Collaboration): First and foremost, I must give credit to my mentor and his peers in the WMHS math department for giving me the idea to have students measure the distances—this is an activity that they have often done with their Trigonometry classes, and I was eager to duplicate it in today’s class. In addition, the classroom management matrix for this lesson (which immediately follows this lesson plan) was similarly designed with the assistance of several faculty members from both within and outside the department. The relatively loose nature of today’s lesson posed unique challenges with respect to classroom management, so I made sure to seek as much input as possible on how to address these challenges. (While this is an academic class consisting primarily of 12th graders, it is nonetheless unreasonable to expect going into the lesson that there will be no management difficulties that arise—though this class did its best to prove me wrong on that count.) Lastly, my mentor and I decided to switch around the plans and add a discussion of the metric system after the measuring activity once students’ lack of familiarity with it became apparent. This lesson made it clear that when done properly, collaboration can be of great benefit to both the students and instructor.


Classroom Management Matrix

Intern: Ben Cohen

School: Winters Mill High School

Date: 2/28/14

Planning for Effective Lesson Management Objective: Solve problems involving applications of triangle area formulas. 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 tape measures passed out while students are solving 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 materials will be prepared prior to the lesson. The longdistance tape measures will be ready for use at the start of class, and all indoor triangles will be marked, if necessary.

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 all problems).

Use the football to give students a chance to volunteer answers to the warm-up and provide thoughts on the Bermuda Triangle handout.

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.

Make sure students know they should ask questions as they arise and not simply when I ask if there are any questions. During the discussion portions of the lesson, have prompts ready to ask the students (e.g., “Do you think it would be feasible for the U.S. to begin a phasein of the metric system?”). Leave 20 minutes at the end of the class for students to finish the post-test. They are aware that this is the plan and that it will not count for a grade, so they can jump right into it at the end rather than needing me to waste time explaining the directions.

All of the problems that will be used in this lesson will be worked out in advanced so I do not have to waste time thinking of how to do them; 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 working on the post-test—do not pass it out earlier; otherwise, students may work on it instead of the post-test since the latter does not count for a grade.

Ensure the indoor triangles (if necessary) are marked in such a manner that all groups will still have sufficient room and traffic through the main hallway is not impeded.

This lesson was designed with student engagement in mind. If there are some students who appear unwilling to share their thoughts or answers during the discussion portion of the lesson, start tossing around the football! 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, given the relatively loose structure of this lesson, behavior management is 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.


Section 5: Instructional Decision-Making


Instructional Decision-Making “Why are there two units on here?” “Why is it marked in feet and centimeters?” Questions like this were rampant following the measuring activity performed during the fifth and final lesson of the unit. As is customary, the instruments students used to measure distances throughout the building were marked in both feet (on one side) and centimeters (on the other). While my mentor and I believed this would be a simple distinction for students to draw— and one with which they would be familiar from their prior courses—it turned out students had virtually no familiarity with the metric system, especially its units of length measurement. A large part of the reason I wanted to expose students to applications of the material in this unit was that the bulk of problems in their textbook asked them to simply solve a given triangle without regard to the units in which the sides are measured. If students are only shown this method of problem solving, however, they will fall into the trap of assuming they are learning all of this material in a vacuum and that it will be of no use in their future pursuits; projects such as the measuring activity ensured this would not be the case. While the focus of this activity was supposed to be the applications of the material, I decided upon discovering students’ lack of familiarity with the metric system that they also would be well served by receiving an introduction to this alternate—and more commonly used—system of measurement. This also helped lead to another class discussion, in this case why the U.S. has not followed the lead of the bulk of the developed world and switched to the metric system. Though not my original intent, this was another useful way of differentiating instruction. This even altered my plans for a subsequent unit. During a lesson on linear and angular speed (part of the unit on radians), I included a lesson on unit conversions. While this is part of the Intermediate Algebra curriculum, I felt students could benefit from a refresher on this, in addition to building on the knowledge of imperial vs. metric measurement introduced in this unit.


Section 6: Analysis of Student Learning


ASSESSMENT DATA COLLECTION STUDENT LEARNING Intern:

Ben Cohen

Year: 2013-14

School: Winters Mill High School

Semester: Spring 2014

Grade:

Subject: Trigonometry/Pre-Calculus

11/12

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 4 0 1 2 0 2 2 2 2 3 2 2 1 2 1.75

10% 20% 40% 0% 10% 20% 0% 20% 20% 20% 20% 30% 20% 20% 10% 20% 18%

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

4 7 10 7 5 6 6 5 7 7 8 9 8 6 5 7 6.69

PERCENT SCORE

CHANGE PRE TO POST (+ or – Percentage Points)

40% 70% 100% 70% 50% 60% 60% 50% 70% 70% 80% 90% 80% 60% 50% 70% 67%

+30% +50% +60% +70% +40% +40% +60% +30% +50% +50% +60% +60% +60% +40% +40% +50% +49%


Analysis of Assessment Data Collection: Student Learning •

In general, it is clear that students learned during this unit, as the class saw an average increase of 49 percent when comparing the pre- and post-test scores. All 16 students saw their scores increase by at least 30 percent; 10 saw an increase of at least 50 percent, including one (H.) whose score increased by 70 percent. Two other students (E. and I.) increased their scores by 60 percent, to 100 percent and 90 percent, respectively.

The class average on the post-test (6.69/10) was slightly lower than I had hoped, but a few low-achieving students dragged down the average. Four students scored 50 percent or lower on the post-test; the remaining 12 students averaged 7.33/10. In addition, as was stated above, all students saw substantial increases in their scores; the two lowest increases (by G. and M., respectively) were still 30-percent gains—to 40 and 50 percent, respectively.

It is somewhat surprising that only two students scored 90 or 100 percent, though two scored 80 percent and five more scored 70 percent. An in-depth review reveals this was due largely to two questions on the post-test that received a total of seven correct answers; this will be discussed in greater detail in the following part of this section. When only the remaining eight questions are considered, the class average jumps from 67 percent to 83 percent, with three students (E., L., and I.) correctly answering all remaining questions and four others (H., B., K., and S.) answering seven out of eight correctly.

Due to the structure of this unit, students were informed that this post-test would not count towards their class grades (they were also given a unit test that did count). A few students’ scores on the post-test diverged wildly from the achievement they demonstrated on the unit test or otherwise suggested a lack of effort—for example, two students (F. and M.) received scores of 50 percent on the post-test due in part to not answering the final three questions—so it is possible the average may have been dragged down by some students not taking this assessment seriously. At the same time, there are also a few students whose scores on the post-test exceeded their scores on the summative assessment; this should also be taken into account when analyzing the scores.

It should be noted that the ESL was performed from February 24-28, which also happened to be the first full week of school in over two months (since December 16-20). A potentially useful exercise would be to compare students’ achievement in this unit to their achievement in other units spread out over more than one week due to snow days and see if they diverge—whether, for example, the frequent disruptions appear to hinder the learning process or may actually be beneficial to students given the increased time to review the material and let it sink in. (Through the week of March 17-21, this was still the only full five-day week of 2014.)


Pretest Item Analysis (Whole Class) Grade: 11/12 Subject: Trigonometry/Pre-Calculus Objectives:

Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5

Unit: Solving Oblique Triangles

Intern: Ben Cohen

Use the Law of Sines to solve oblique triangles given AAS, ASA, or SSA. Use the Law of Cosines to solve oblique triangles given two sides and the included angle (SAS). Use the Law of Cosines to solve oblique triangles given all three side measures (SSS). Find the area of an oblique triangle using Heron’s Formula given SSS. Find the area of an oblique triangle given SAS, ASA, or AAS; solve application problems. PRETEST ASSESSMENT ITEMS

1

2

3

STUDENT

2 3

5

6

7

8

9

10

5

1

2

ALIGNMENT WITH LESSON OBJECTIVES

2 1

4

X X X

1

1

3

X X

5

4

4

X

X

4

X

5 6

X

X

7

X X X

8 9 10 11 12

X X

13 14 15 16 TOTAL CORRECT RESPONSES

X X 8

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

X X X X X X 8

X X X X

X 0

0

6

0

0

4

2

0

TOTAL CORRECT RESPONSES

1 2 4 0 1 2 0 2 2 2 2 3 2 2 1 2 28


Post Test Item Analysis (Whole Class) Grade: 11/12 Subject: Trigonometry/Pre-Calculus Objectives:

Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5

Unit: Solving Oblique Triangles

Intern: Ben Cohen

Use the Law of Sines to solve oblique triangles given AAS, ASA, or SSA. Use the Law of Cosines to solve oblique triangles given two sides and the included angle (SAS). Use the Law of Cosines to solve oblique triangles given all three side measures (SSS). Find the area of an oblique triangle using Heron’s Formula given SSS. Find the area of an oblique triangle given SAS, ASA, or AAS; solve application problems. POST TEST ASSESSMENT ITEMS

1

2

3

STUDENT

1

1

X X X X

X X X X X X

1 2 4

X X

5 6 7

X X

9

X

11 12 13 14

X X X X X X

15 16 TOTAL CORRECT RESPONSES

X 11

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

3

X X X

X

8 10

5

6

7

8

9

10

5

5

2

ALIGNMENT WITH LESSON OBJECTIVES

2

3

4

X X X X X X X 12

X X X X X X 13

X X X X X X X 10

5

4

4

X X X X X X X X X X X X X

X X X X X X X X X X X X X X X X 16

X X X X

X 14

X X X X X X X X X X X 15

X X

X

X X

X X

X

X X X X X

X

3

X

X

4

9

TOTAL CORRECT RESPONSES

4 7 10 7 5 6 6 5 7 7 8 9 8 6 5 7 107


Discussion of Post Test Item Analysis (Whole Class) •

With the exception of question 8 (assessing students’ ability to solve an application problem)—which saw one fewer correct answer on the post-test than on the pretest—all questions on the post-test received more correct responses than their counterparts on the pretest, often by substantial margins.

Questions 6 and 7, which pertained to the lesson 4 objective (finding the area of an oblique triangle using Heron’s formula), saw the largest increase: While neither received a single correct answer on the pretest, all 16 students correctly answered question 6 on the post-test, and 15 out of 16 students correctly answered question 7. In some respects, this is not surprising—since these were free-response questions, students’ ability to correctly guess on the pretest was greatly limited, and since this is one of the simpler objectives from the unit, it is not shocking that students met with such success.

Question 3, which asked students to use the Law of Sines to solve an oblique triangle, jumped from 0 correct answers on the pretest to 13 on the post-test. The lack of a single correct answer on the pretest is almost certainly due to the structure of the question—as a free-response rather than multiple-choice question, it would be extremely difficult for students to correctly guess. That 13 out of 16 students correctly answered suggests general mastery of this topic. (Question 2, which also evaluates students’ knowledge of the Law of Sines, received 12 correct answers on the post-test, though eight students correctly answered question 2 on the pretest.)

Question 4, which asks students to use the Law of Cosines to solve an SSS triangle, also saw a substantial increase in correct answers, going from none on the pretest to 10 on the post-test. Again, since this is a free-response question, it is highly unlikely any students would have been able to correctly guess on the pretest like with a multiple-choice question; unlike the lesson 4 objective, however, solving a triangle given SSS was arguably the most challenging of the five objectives, so it is gratifying to see 10 out of 16 students were able to solve this problem.

Fourteen students correctly answered question 5, which concerned finding the area of an oblique triangle using the sine formula; this is an increase from six on the pretest. It should be noted, however, that question 8 was on the same topic but received fewer correct answers (three on the post-test, compared to four on the pretest). The reason for this is likely that students had not covered application problems involving extra steps similar to question 8; it appears they should not have been expected to make this leap on their own. (The low rate of correct responses to question 9—four, up from two on the pretest—is likely attributable to this as well.)

Question 1 received 11 correct answers, up from eight on the pretest; this involved solving a triangle given SAS. This proved to be a frequent sticking point for students during the unit, though the increase in correct responses is encouraging. (Question 10—a free-response word problem on the same topic—received no correct answers on the pretest but nine on the post-test.)


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

STUDENT LEARNING Intern:

Ben Cohen

Year: 2013-14

School: Winters Mill High School

Semester: Spring 2014

Grade:

Unit: Solving Oblique Triangles

11/12

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

2 0 1 2 2 2 2 3 2

20% 0% 10% 20% 20% 20% 20% 30% 20%

10 10 10 10 10 10 10 10 10

7 7 5 6 5 7 8 9 6

70% 70% 50% 60% 50% 70% 80% 90% 60%

+50% +70% +40% +40% +30% +50% +60% +60% +40%

1.78

18%

6.67

67%

+49%


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 49 percent from pre- to post-test, identical to the 49 percent average for the class as a whole. All nine students saw their scores increase by at least 30 percent, with five experiencing an increase of 50 percent or greater, including one student (H.) who went from a 0 on the pretest to a 70 percent on the post-test.

As with the class as a whole, the average for this subgroup on the post-test (6.67/10) was lower than I had hoped, although the female students’ average was virtually identical to that of the class as a whole (6.69/10). While all students scored at least 50 percent on the post-test, only two (L. and I.) scored 80 percent or higher—in the case of L., this is particularly impressive since she missed the first two days of the unit due to illness.

The only two students in the class who correctly answered question 9 (N. and I.) are members of this subgroup, as are two of the three students who correctly answered question 8 (C. and A.). Nevertheless, when these two questions are removed from the analysis and only students’ scores on questions 1-7 and 10 are considered, the subgroup average increases to 70 percent, which is an improvement but still below the adjusted average of 83 percent for the full class.

The female students’ average was virtually identical to that of their male peers (6.67/10 vs. 6.71/10); while the only student to score 100 percent (E.) is a male, the secondhighest score (90 percent) was attained by I., a member of this subgroup. In addition, one of the two students who scored an 80 percent (L.) is a member of this subgroup, as are three of the five students to score 70 percent (C., H., and B.). It is extremely gratifying to see the female students scoring on par with their male peers, particularly given that this was not the case with the female students during my fall ESL.

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, five scored in the “upper tier” (seven or more out of 10), two were in the “middle tier” (six out of 10), and two were in the “lower tier” (five out of 10). This is actually a better result than had been predicted by students’ grades (three grades of “A,” five of “B,” and one of “C”).

Upon analyzing students’ interim grades (following this unit), it became clear that the gender gap in both this class and the second-period section was virtually nonexistent— there were no discernable patterns of female students scoring lower than their male peers, and in fact the lowest-achieving students in both sections are all males. This 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 whether it is possible to duplicate these factors in future classes.


DISAGGREGATED ASSESSMENT DATA COLLECTION – STUDENTS BELOW 90% ATTENDANCE (Name of Subgroup)

STUDENT LEARNING Intern:

Ben Cohen

Year: 2013-14

School: Winters Mill High School

Semester: Spring 2014

Grade:

Unit: Solving Oblique Triangles

11/12

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 Subgroup Average

10 10 10 10 10 10

1 2 0 1 2 2

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

10 10 10 10 10 10

4 7 7 5 5 7

40% 70% 70% 50% 50% 70%

+30% +50% +70% +40% +30% +50%

1.33

13%

5.83

58%

+45%


Analysis of Disaggregated Data – Students Below 90% Attendance: Student Learning •

As was the case for the whole class, it is clear that students in this subgroup learned during this unit. These students saw an average increase in 45 percent from their pre- to post-test scores, just below the full-class average of 49 percent. However, it should also be noted that these students had a very low average on the pretest to begin with—no student scored above 20 percent, and one (H.) scored a 0. Thus, this is in part a function of students’ low initial achievement, though several students did score very well on the post-test in addition.

This subgroup had a rather low average on the post-test: 5.83/10, which was significantly lower than the averages for female students (6.67/20) and the full class (6.69/10). Like the full class, these students fared poorly on questions 8 and 9 (only one student—C.— correctly answered question 8, and none correctly answered question 9); however, they also did not score well on questions 10 (only C. answered correctly) and 1 (H. and S. provided the only correct answers), which stands in sharp contrast to their peers.

C. is one of only three students in the class who correctly answered question 8, so this strong achievement should not be discounted. Nevertheless, when questions 8 and 9 are removed from the analysis and only students’ scores on questions 1-7 and 10 are considered, the subgroup average increases to 71 percent, which is a significant improvement but still below the adjusted average of 83 percent for the full class. (If question 10 is also excluded, the average increases to 79 percent—all remain questions other than question 2 received at least four correct answers, out of six students.)

In identifying this subgroup, I stated I felt the students in this group were likely to achieve at a level below their peers, since mathematics is a sequential subject and they had missed at least 10 percent of the instructional time so far this semester, meaning they would not have as strong a foundation as the other students in the class. The low achievement on the post-test certainly appears to confirm this hypothesis; no student scored higher than 7 out of 10 on the post-test, and the student with the lowest score (G., 4/10) is a member of this subgroup. It should be noted that three of the four questions on which the students in this group did particularly poorly are questions that involve multistep problem solving, which gave all students in the class trouble. However, these students missed not only content instruction but also valuable problem-solving techniques during their myriad absences; thus, while the material in this unit did not necessarily directly build on past concepts, they were still at a deficiency in this skill set, which was a prerequisite for success in this unit.

Ironically, five of the six students in this subgroup had perfect attendance during the unit (M. missed the first two days due to illness; she scored a 50 percent on the post-test). I would be interested in doing a similar study when multiple students have been absent for two or more days of a given unit to see how their achievement on the summative assessment compares to their peers.


Pretest Item Analysis Disaggregated Subgroup – Female Students Grade: 11/12 Subject: Trigonometry/Pre-Calculus Objectives:

Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5

Unit: Solving Oblique Triangles

Intern: Ben Cohen

Use the Law of Sines to solve oblique triangles given AAS, ASA, or SSA. Use the Law of Cosines to solve oblique triangles given two sides and the included angle (SAS). Use the Law of Cosines to solve oblique triangles given all three side measures (SSS). Find the area of an oblique triangle using Heron’s Formula given SSS. Find the area of an oblique triangle given SAS, ASA, or AAS; solve application problems. PRETEST ASSESSMENT ITEMS

1

2

3

2

1

1

X

X

STUDENT 1

4

5

6

7

8

9

10

5

1

2

ALIGNMENT WITH LESSON OBJECTIVES

3

5

4

4

2

X

3 4

X

X X X

5 6 7 8

X X

9 TOTAL CORRECT RESPONSES

4

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

X X X 4

X X X

0

0

X 4

0

0

3

1

0

TOTAL CORRECT RESPONSES

2 0 1 2 2 2 2 3 2 16


Post Test Item Analysis Data for Disaggregated Subgroup – Female Students Grade: 11/12 Subject: Trigonometry/Pre-Calculus Objectives:

Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5

Unit: Solving Oblique Triangles

Intern: Ben Cohen

Use the Law of Sines to solve oblique triangles given AAS, ASA, or SSA. Use the Law of Cosines to solve oblique triangles given two sides and the included angle (SAS). Use the Law of Cosines to solve oblique triangles given all three side measures (SSS). Find the area of an oblique triangle using Heron’s Formula given SSS. Find the area of an oblique triangle given SAS, ASA, or AAS; solve application problems. POST TEST ASSESSMENT ITEMS

1

2

3

2

1

1

X

X X X

X X X X X

STUDENT

3 4

X

5 6 7 8 9 TOTAL CORRECT RESPONSES

5

6

7

8

9

10

5

2

ALIGNMENT WITH LESSON OBJECTIVES

1 2

4

X X X X 6

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

X X X X 7

X X X 8

3 X X X X X X 6

5

4

4

5

X X X X X X X X

X X X X X X X X X 9

X X

X

8

X X X X X X 8

X X

X 2

X

X X X

2

4

TOTAL CORRECT RESPONSES

7 7 5 6 5 7 8 9 6 60


Discussion of Post Test Item Analysis: Female Students •

With the exception of question 8 (an application problem involving the area of an oblique triangle), which saw one fewer correct answer, all questions on the post-test received more correct answers than their corresponding questions on the pretest, many by substantial margins, suggesting that student learning occurred in this unit.

The most significant improvement occurred on questions 6 and 7, which asked students to use Heron’s formula to find the area of an oblique triangle. Neither question received a single correct response from students in this subgroup on the pretest, but all nine students correctly answered question 6 on the post-test, and eight out of nine solved question 7. These success rates mirrored those of the full class; all 16 students correctly answered question 6, and the only student who incorrectly answered question 7 (F.) is a member of the female-student subgroup.

Students in this subgroup were also highly successful at questions assessing the objective from lesson 1 (the Law of Sines); no student correctly answered question 3 on the pretest, but eight out of nine (89 percent; all except B.) correctly answered it on the post-test. (This is a slightly higher percentage than for the full class: 81 percent, or 13 out of 16 students.) Seven out of nine students in this subgroup (78 percent) correctly answered question 2, compared to four out of nine on the pretest; this is again slightly above the percentage for the full class (12 out of 16; 75 percent).

Six out of the nine students (67 percent) were able to solve an oblique triangle given all three side measures; no students were able to do this on the pretest. This is again in line with the full-class percentage (63 percent) and also closely mirrors the gains for the class as a whole, which also saw no correct answers to question 4 on the pretest.

Question 1, which asked students to use the Law of Cosines to solve an SAS triangle, also received six correct responses, up from four on the pretest. Interestingly, one student (C.) correctly answered question 1 on the pretest but not on the post-test, suggesting an initial lucky guess. This is slightly below the full-class percentage of correct responses (11/16, or 69 percent), though still quite close. The other three students who answered this question correctly on the pretest (N., L., and I.) also answered it correctly on the post-test.

Eight out of nine students (all except A.) were able to find the area of an oblique triangle using the sine formula; this represents an increase from four on the pretest. (Interestingly, A. did answer this question correctly on the pretest, again suggesting a correct guess rather than content knowledge.) Questions 8 and 9, however, saw more limited success. Question 8 actually received fewer correct responses than on the pretest (three students correctly responded on the pretest; two different students correctly responded on the posttest). Two students correctly responded to question 9, and four to question 10 (compared to none on the pretest). As with the full class, the data suggest that the instructional time devoted to these multi-step application problems was insufficient.


Pretest Item Analysis Disaggregated Subgroup – Students Below 90% Attendance Grade: 11/12 Subject: Trigonometry/Pre-Calculus Objectives:

Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5

Unit: Solving Oblique Triangles

Intern: Ben Cohen

Use the Law of Sines to solve oblique triangles given AAS, ASA, or SSA. Use the Law of Cosines to solve oblique triangles given two sides and the included angle (SAS). Use the Law of Cosines to solve oblique triangles given all three side measures (SSS). Find the area of an oblique triangle using Heron’s Formula given SSS. Find the area of an oblique triangle given SAS, ASA, or AAS; solve application problems. PRETEST ASSESSMENT ITEMS

1

2

3

2

1

1

X X

X

STUDENT 1 2

4

5

6

7

8

9

10

5

1

2

ALIGNMENT WITH LESSON OBJECTIVES

3

5

4

4

3

X

4

X

5 6 TOTAL CORRECT RESPONSES

X 3

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

X 2

0

0

1

X 0

0

1

1

0

TOTAL CORRECT RESPONSES

1 2 0 1 2 2 8


Post Test Item Analysis Data for Disaggregated Subgroup – Students Below 90% Attendance Grade: 11/12 Subject: Trigonometry/Pre-Calculus Objectives:

Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5

Unit: Solving Oblique Triangles

Intern: Ben Cohen

Use the Law of Sines to solve oblique triangles given AAS, ASA, or SSA. Use the Law of Cosines to solve oblique triangles given two sides and the included angle (SAS). Use the Law of Cosines to solve oblique triangles given all three side measures (SSS). Find the area of an oblique triangle using Heron’s Formula given SSS. Find the area of an oblique triangle given SAS, ASA, or AAS; solve application problems. POST TEST ASSESSMENT ITEMS

1

2

3

2

1

1

X

X X X

STUDENT

2 4 5 6 TOTAL CORRECT RESPONSES

5

6

7

8

9

10

5

2

ALIGNMENT WITH LESSON OBJECTIVES

1 3

4

X 2

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

X 4

X X X X X X 6

3

5

4

4

5

X X X X X X 6

X X X

X

X X X X 4

X X X X X X 6

X X 5

1

X

0

1

TOTAL CORRECT RESPONSES

4 7 7 5 5 7 35


Discussion of Post Test Item Analysis: Students Below 90% Attendance •

Somewhat surprisingly, three questions on the post-test saw either no change or a reduction in the number of correct answers: questions 1 (three correct on pretest, two on post-test), 8 (one correct on pre- and post-tests), and 9 (one correct on pretest, none correct on post-test). This seems to lend credence to the theory that these students’ poor attendance would eventually catch up with them and inhibit their ability to succeed in future units.

All other questions did see an increase in the number of correct answers—in some cases, a substantial one. Question 3, for example (using the Law of Sines to solve a triangle), received no correct answers on the pretest, but all six students correctly answered it on the post-test. Question 2, which was also aligned with this objective, went from two to four correct answers. (It should also be noted that question 2 was multiple-choice, while question 3 was free-response, so these patterns are the opposite of what was observed for other subgroups and the full class.)

Question 6, which asked students to find the area of a triangle using Heron’s formula, also went from no correct answers on the pretest to perfection on the post-test. Question 7 experienced a similarly sharp rise; five out of six students (all except F.) correctly answered it on the post-test, and this student was also the only one in the class to not answer it correctly.

Question 4, which asked students to use the Law of Cosines to solve a triangle given SSS, jumped from zero correct answers on the pretest to four on the post-test. This was the fourth question to see an increase in correct answers of 67 percent or more among this subgroup, joining questions 3, 6, and 7.

Student success with application problems (objective 5) was more mixed. All six students correctly answered question 5—compared to one on the pretest—but only one student correctly answered question 8 (the same as on the pretest), and question 9 went from one correct answer on the pretest to none on the post-test. Again, this suggests that there was insufficient instruction on these types of problems; students are aware of the mathematics behind the problems, but the additional steps that must be taken on these problems added complications that made it difficult for students to solve the problems.

Question 10 also saw greatly limited success, presumably for the same reasons as questions 8 and 9. While students are aware of how to use the Law of Cosines to find the missing side given SAS, the additional steps beyond this that needed to be completed for this particular question proved to be too much for them to handle. In the future, I will be sure to explicitly address these during class, rather than relying on students to make the jump.


Section 7: Reflection and Self-Evaluation


Reflection and Self-Evaluation In this unit, the objective at which students were the most successful was the lesson four objective: “Students will be able to use Heron’s formula to find the area of an oblique triangle given all three side lengths.” A total of 31 out of 32 students responses to the two questions on the post-test concerning Heron’s formula (questions 6 and 7) were correct, a total of 97 percent—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 my mentor’s suggested activity for the final class, which asked students to measure the sides of and find the area of a series of oblique triangles around the building. This was a very helpful, hands-on activity that ensured students were truly actively engaged in the learning process, and it certainly paid dividends, as evidenced by the near-universal rate of correct responses. 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. It is also possible the activities I had planned at the start of the fourth lesson helped with students’ understanding of these concepts. My mentor has always introduced a historical connection when covering this unit and generally has students read a bio of Heron of Alexandria (for whom the formula is named); while I was not able to use this exact strategy, I did include a substantial cross-curricular component, asking students to read a series of passages exposing them to the historical background of this topic as well as others they have studied throughout their mathematical careers. Student reactions to the activity were decidedly mixed, but it does appear that this aided the learning process. To be frank, it should also be noted that this is arguably the easiest topic in the unit. Whereas most of the other problem-solving techniques require an algebraic manipulation to set up the correct equation, Heron’s formula requires little more than substituting the given numbers


into a formula. To be fair, there are also several steps to solving a problem using Heron’s formula, which means there are ample opportunities for students to err, as one student did; just the same, I felt going into the unit that this would be one of the easier topics for students to grasp, and the post-test data seem to confirm that conclusion. Lastly, while a handful of students struggled in this unit due to a lack of command of the definitions of sine and cosine and their respective signs, this knowledge is not necessary to determine the area of an oblique triangle, meaning these students were not put at a disadvantage. On a Law of Cosines problem, for example, students need to remember to solve for the missing angle values in the correct order (largest first if using inverse cosine; smallest first if using inverse sine). Heron’s formula, however, presents no such complications, since it only requires the side lengths, and no angle measures, as input. While this certainly 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 as well as reinforced over the course of this unit include: Organization and preparation are crucial. It has become clear to me over the course of this placement 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 (a particular concern this year), 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 (applications of oblique triangles) was pushed back one day to allow students an additional day of mixed practice with the Laws of Sines and Cosines. 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. In addition, the administration had previously hinted that there would be an emergency drill this week; while it never materialized, it did remind me of the importance of planning for the unexpected and being able to adjust as necessary. Don’t be afraid to use the resources around you. Prior to the final lesson of the unit, my mentor suggested I consider using an activity he had previously used several times, to great success—having students measure the distance between objects of significance outside of the school building, and then determining the area of the resulting oblique triangles. While it was too cold to bring students outside, forcing us to come up with three triangles inside the building, this was nonetheless a very helpful activity; students thoroughly enjoyed it, and it also ensured they were active participants in class rather than spectators. I also collaborated with a handful of staff both within and outside the math department during the construction of the classroom management matrix that accompanied this lesson plan. Don’t make assumptions about what students have retained from previous classes. As I stated in the respective lesson plans, students at this level are expected to be able to perform a series of algebraic manipulations, having already taken Intermediate Algebra and Algebra II. However, it became clear to me that they were struggling with these manipulations during our


derivation of the Law of Cosines; I had assumed (erroneously) that they would be able to perform these manipulations since they had just taken Algebra II the previous year or previous semester. What I failed to take into account is that the use of a trigonometric identity (cos C) rather than a simple x was likely to lead to significant confusion, and that is exactly what happened. I made sure to include an informal assessment during the first lesson to see how much knowledge students had retained from Geometry; I should have done the same for their two algebra classes. Make sure to expose students to applications frequently. The most common refrain I have heard from students throughout my time at WMHS is, “When are we ever going to use this?” I made sure to begin each class with an application problem—whether it was finding the area of the Bermuda Triangle or the distance a fielded bunt must be thrown to first base—given the relative dryness of this material. While this did take up a not-insignificant amount of instructional time, it also exposed students to the awesome power of the material in the unit, as well as the significant role it played throughout history. My goal is always to ensure that students are not learning these mathematical concepts in a vacuum but are instead aware of the roles they play in their everyday lives, and constant, gradual exposure to these applications goes a long way towards satisfying that goal. Technology can be a boon—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. Even when well intentioned, sometimes the integration of technology can backfire, as happened when I attempted to use a Geometer’s Sketchpad demonstration to derive the Law of Cosines. It soon became clear that the mathematics being discussed in this demonstration were at a level too


abstract for students, and they were clearly struggling to grasp the concepts. I ended up effectively scrapping the demonstration and running through an algebraic derivation, which, while low-tech, was also significantly easier for students to understand. 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. Students can benefit greatly from working with their peers. 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 continue to include opportunities for one-on-one or group work (including student-to-student activities), such as the reading activities and discussion from the four lesson, throughout the lessons in future units. This last insight segues into one professional development goal that emerged from the ESL, that being the integration of reading and writing into the mathematics curriculum. Because of the nature of its material, it is very difficult to properly integrate these components into a math class—merely having students read about solving a problem, for example, will likely leave them with insufficient knowledge of how to solve it. In addition, many peer-review journal articles are written at an extremely high level (by and for individuals with Ph.D.’s) and will thus be virtually


impossible for students to comprehend. Simpler reading and writing activities—such as the Bermuda Triangle article distributed at the start of the fifth class, which, in retrospect, should have been the focus of the fourth lesson’s activities—can certainly help enhance the educational experience for students, as could research projects—for example, I could have brought class to the media center one day and had them research one historic use of oblique triangles (such as land surveying). Problems that require written explanation—for example, “Explain why it is possible to determine whether an angle is oblique by using the Law of Cosines, but not the Law of Sines”—can also significantly help this pursuit, in addition to exposing students to the thinking they will be expected to perform following the switch to Common Core. Another area that I have continued to make a priority with respect to to professional development is the integration of technology. As I detailed above, instructional technology can be an invaluable resource—if it is deployed properly. During the National Council of Teachers of Mathematics conference last October at the Baltimore Convention Center, I had the opportunity to attend a workshop led by the creator of Geometer’s Sketchpad that focused on its integration throughout the math curriculum; I made copious use of many of the strategies discussed at this workshop throughout the unit, including creating interactive, animated demonstrations of the derivations of the Laws of Sines and Cosines. There also exist numerous other avenues for the integration of technology, such as the TI-Nspire calculators in the classroom, which I am planning to make the focal points of some calculator-heavy lessons in Algebra I later this semester. I have realized that for full effectiveness, technology integration should take multiple forms throughout the semester. 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 this critical field.

Evidence of Student Learning - Spring 2014  
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