COE 11: Use of Technology

Rationale

My chosen artifact for this standard is the special acute angles lesson plan from

Trigonometry/Pre-Calculus at Winters Mill High School. In this lesson, students were

introduced to the trigonometric values for the three special acute angles (30o, 45o, and 60o), as

well as how to use them as “reference angles” when dealing with angles that had a multiple of

90o added to these numbers. This lesson focused on the derivation of these values, using students’

existing knowledge from Geometry (such as the side lengths of 30-60-90 and 45-45-90 right

triangles) and the definitions of sine, cosine, and tangent. (Demonstrations from two additional

lessons—introduction to radians and linear and angular speed—are included as well.)

This lesson illustrates standard 11 because it is a specific example of the integration of

technology to enhance students’ understanding of the content and make it accessible for all

students regardless of learning style. I wanted to tie these derivations to students’ existing

knowledge, but I wanted something more interactive and dynamic than a freehand demonstration

on the whiteboard. Thanks to Sketchpad, I was able to include components to the lesson that

would not have been possible with a whiteboard-based demonstration, such as shading in one of

the two triangles created after bisecting an angle or providing instant updates of the values of

sides and angles in the newly created triangles. The focus in this lesson was on ensuring students

understood where these trigonometric values came from, not just what they were, and the

integration of technology made it possible to clearly and efficiently demonstrate this.

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.

This activity had a strong positive effect on student learning. The best example of this is

given by the surveys that I asked students to fill out at the end of my placement—several

students specifically mentioned my use of technology as one of the most effective instructional

strategies I used, including one who credited these demonstrations for making this the “first year

I actually understand math.”

This artifact closely parallels the “planning instruction” and “teaching” portions of the

CPTAAR cycle. As I have always said, technology for technology’s sake does not improve

outcomes for students; rather, the technological component needs to either provide a means of

making the topic more accessible for students or allowing them to extend their knowledge (such

as focusing on the why and how rather than just the what). In this instance, the mere fact that I

had a computer program that let me plot a triangle in the coordinate plane, rather than simply

drawing it on the board, didn’t improve students’ learning—the instruction that specifically used

this program’s capabilities to demonstrate the derivations of the trigonometric values did.

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

Unit The Trigonometric Functions

Topic Acute and Reference Angles

Date January 31, 2014

Grade 11/12

Class Size 12

Time 9:35-10:55 a.m.

Mentor Mark Heffner

School Winters Mill High School

Intern Ben Cohen

Standard(s): ____ CCSS Other: Maryland Content Standards 2.1.2.1 The student will locate points on the polar coordinate system, including points with rotation greater than 360o. 2.2.2 The student will solve problems using two-dimensional figures and/or right-triangle trigonometry.

Big Idea or Essential Question Yesterday, students focused on the special cases in which an angle is determined by a point on the x- or y-axis (the cases of quadrantal angles). Today, students will focus on three additional special cases: 30-degree, 45-degree, and 60-degree angles. (Students likely have some familiarity with these angles from Geometry, considering the significance of 30-60-90 and 45-45-90 right triangles). Their evaluations will be derived through the use of Geometer’s Sketchpad to ensure students understand not only what the values are, but also how they are derived. In addition, today’s lesson will teach students how use these angles as reference angles, which allow students to determine the trigonometric values of obtuse angles that are simply these special acute angles with a multiple of 90 degrees added. This will include a lesson on the importance of knowing whether these values are positive or negative in a given quadrant, as well as the “all students take calculus” mnemonic for remembering this.

Alignment with Summative Assessment Questions 9, 11, and 12 on the CCPS summative assessment asks students to find a trigonometric value (secant, cosine, and cotangent, respectively) of an angle coterminal with one of the special acute angles.

Lesson Objective Students will understand how to derive and identify the values of the six trigonometric functions for special acute angles (30o, 45o, and 60o). They will also successfully use reference angles to find these values for angles greater than 90o.

Formative Assessment Students will be asked to complete six problems in the book for classwork; they will also complete an additional 10 problems for homework. All problems will ask them to provide the six trigonometric function values for angles coterminal with special acute angles. Informal assessments are integrated throughout the lesson; remediation of Algebra II material (such as the definitions of the four quadrants of the coordinate plane) will be built into the lesson if 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 ►►►►► Instructional Decisions Based on (What prior knowledge of learners are you Knowledge of Learners using to plan this lesson?

DATA

Formal data - Pretest, formative assessment, checklists, etc. • All students in the class successfully completed last night’s homework, which asked them to find the values of trigonometric functions for angles coterminal with the four quadrantal angles.

RESPONSE TO DATA (required)

• This shows that students already have a solid grasp of the fundamentals they will need to succeed in today’s lesson, which utilizes an extremely similar skill set. Although I am still prepared to include a remediation component if necessary—especially since it is very early in the semester—the expectation is that students will be able to move straight into the material for this lesson.

Informal data - Observations of students, reflections from previous lesson(s), anecdotal records, etc. • J. has just added the class after moving down from the Honors section. Today is her second meeting with this section. • Students should know which quadrants of the coordinate plane correspond to positive and negative x- and y-coordinates.

• As she has already been in a section of the class that is more advanced and moves at a faster pace than this one, lack of familiarity with the material will likely not be a problem. She will need to get caught up on the work she has missed, however; I will work with her during FLEX time to facilitate this. In addition, there may be a transition period since she is used to honors-level pacing; however, as I also had her in standard-level Algebra II last semester, she is used to my teaching style and so this should also not be a major issue. • This does not mean that I do not need to introduce these definitions, but I may be able to make this portion of the lesson more discovery-focused based on their prerequisite knowledge. The structure and time of this portion of the lesson will be adjusted based on this.

Individual or Small Group Needs Specific ►►►►► Differentiated Practices Specific to this Lesson to this Lesson (Ex. IEP/504 accommodations, ELL, social concerns, (Instruction and/or Assessment) gifted/talented, multicultural/equity measures, etc.) • D. has a 504 plan that allows him access to a scribe and alternate test-taking arrangements.

• I must ensure he has access to a copy of the notes from today’s lesson if he needs it rather than forcing him to copy everything down and potentially tire out his arm. If necessary, I will work with the Falcon Learning Lab staff to provide accommodations for today’s quiz.

• 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 TI-84…”

III. Instructional Procedures – What instructional strategies will I use to ensure that every child is a successful learner?

Instructional Materials/Resources Students will be asked to complete six problems in the book for classwork; they will also complete an additional 10 problems for homework. All problems will ask them to provide the six trigonometric function values for angles coterminal with special acute angles. Students will also fill in a separate worksheet with the values of special acute angles (30o, 45o, and 60o). This lesson will also use a demonstration in Geometer’s Sketchpad to illustrate the derivation of these values. Technology Integration This lesson will utilize a demonstration in Geometer’s Sketchpad to help students derive the values of trigonometric functions for special acute angles, using right triangle trigonometry (30-60-90 and 45-45-90 right triangles). A portion of the lesson will also be devoted to techniques for finding their reciprocal functions on TI-83 and TI-84 calculators for those students who utilize these devices, as well as on the classroom’s TI-Nspire calculators. Cross-curricular Connections This lesson will lay the groundwork for applications to which students will be exposed in subsequent lessons throughout not just this unit, but the entire class. As this is a foundational lesson, the focus is on equipping students with adequate scaffolding to ensure they can succeed in the rest of the class, so there will be no explicit cross-curricular connections in this lesson so that the full amount of instructional time can be devoted to building a sound foundation. There are countless cross-curricular connections throughout this class, all of which will make use of the background knowledge introduced today. 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. • Opportunities for student engagement are included throughout the lesson. Students will be asked to provide answers to questions asking for the conversion of polar coordinates into rectangular form; they will also be asked to provide the basic form of an equation for a line and a circle as well as the values of sine, cosine, and tangent for specified angles. • 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

Approximate Time Procedure

Planned Beginning

• Warm-up • Motivation • Bridge

5 minutes

• For the warm-up, students will be asked to find the six trigonometric values for an angle of 540o, which is coterminal with 180o. This will assess both their knowledge of the quadrantal trigonometric values introduced in the previous lesson and their understanding that these values are identical for coterminal angles. • In the previous lesson, students derived values of the trigonometric functions for the quadrantal angles (0o, 90o, 180o, and 270o). Today, they will focus on other special acute angles that recur throughout trigonometry: 30o, 45o, and 60o.

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 by finishing up the final two examples from yesterday’s lesson: finding the other five values of the trigonometric functions given one and the quadrant in which the angle terminates. After this, ask if there are any questions about this topic. • Give students five minutes to review the quadrantal angle values from the previous class. After this, they will take a quiz on these values. They will have at most 10 minutes to complete the quiz. • Following the quiz, introduce students to the methods for finding the trigonometric values of acute angles: 30o, 45o, and 60o. These angles recur throughout geometry and trigonometry and so have a special significance. • Begin by deriving the values for 60o, using Geometer’s Sketchpad to demonstrate. Start with an equilateral triangle with a side length of 2 and three angle measures of 60o. An angle bisector that is perpendicular to the x-axis creates two distinct 30-60-90 right triangles; we know the length of r is 2 (unchanged) and the length of x is 1, so we can use this information to solve for y. With all three of these values known, we can find the values of the six trigonometric functions for the angle. • What about for 30o? This time (again using Geometer’s Sketchpad), construct an angle bisector that passes through the origin. This again creates two 30-60-90 right triangles, but this time x, y, and r are relative to a 30o angle. In addition, y and r are known; we can use these to solve for x, and then use these three values to evaluate the trigonometric functions. • The final case is a 45-45-90 right triangle. This is the simplest case; we can construct such a triangle with lengths of 1 for x and y, use these to find r, and then use this information to evaluate the

six trigonometric functions. Illustrate this derivation for students using Geometer’s Sketchpad as well. • Have students fill in their charts with the six values for these three acute angles.

Enrichment or Remediation (As appropriate to lesson)

30 minutes

• We have derived the values of the trigonometric functions for acute angles, but what about angles greater than 90o? • Explain to students that every non-quadrantal angle has what is called a reference angle, which is the positive acute angle made by the terminal side of the angle in question and the x-axis. • To find the values of the trigonometric functions for an angle greater than 90o, simply find these values for its reference angle, and then verify that the sign is correct. For example, to find the sine of 210o, take the sine of the reference angle of 30o; since it is located in quadrant III, the sine will also be negative. • Go through 3-4 examples, using arbitrarily selected angles. If time permits, use this to derive the remaining nine values of the trigonometric functions for special angles (120o, 135o, 150o, etc.). • Have students complete six problems from the book for classwork. This will be collected and scored. • Circulate as students are working to answer any questions and informally assess students’ understanding of the material.

Assessment/ Evaluation

Ongoing

• Students will be informally assessed throughout the lesson. During the lecture component, the instructor will look around the room to see if any students are visibly having difficulty understanding. As students are working individually on the book problems, the instructor will circulate to answer questions and ensure all students comprehend the material and are remaining on topic. • The warm-up is also designed to serve as an informal assessment; the instruction will be adjusted based on whether remediation is necessary or whether students are comfortable enough to proceed. • Take time throughout the lesson to ask if any students have questions about the material.

Planned Ending (Closure)

• Summary • Homework

5 minutes

• Briefly summarize and restate the information that was covered during the lesson. • Pass out homework (worksheet with 10 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?

• One hundred percent of students were able to find the correct absolute value of angles between 0o and 360o. However, three out of the 12 students consistently gave only positive values, rather than the appropriate negative values for Quadrants II, III, and IV. This lesson focused primarily on ensuring students understood the derivations of the three special acute angles; in light of the assessment data, I will extend this topic to a second day to ensure the sign/quadrant relationship has been adequately covered. In retrospect, I should have planned to spread out the topic over two days due to the focus on derivations today. • All 12 students volunteered to provide answers to the properties of 30-60-90 and 45-45-90 right triangles as I was going through the Geometer’s Sketchpad demonstration. Again, while the primary purpose of the demonstration was to link the topic to students’ prior knowledge, the amount of knowledge retained by students suggests they should encounter minimal difficulties with this 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?

• The Geometer’s Sketchpad proved to be quite effective—both the students and my mentor remarked on how helpful it was after class. It should be noted that these technological derivations were designed to supplement students’ knowledge—this unit requires only that students commit these values to memory. Still, I felt it was important that students know not only the values, but also where they came from, particularly as it was linked to their previous knowledge from geometry class of 30-60-90 and 45-45-90 right triangles. Especially for an advanced class, I felt it was key to go through this derivation instead of simply giving students the values for the special acute angles. • While the Sketchpad derivation was very effective, I should have spent less time on the derivation of the relationship between sign of the trigonometric function and quadrant (e.g., cosine is negative in Quadrants II and III). Certainly, it is not unreasonable to expect that students in this class would recall for which quadrants the values of x and y are positive and negative, but for this derivation, I should have guided students to the answer using their existing knowledge, instead of simply expecting them to combine their existing knowledge about this with the definitions of sine, cosine, and tangent to determine the signs of functions in each quadrant. • The most common mistake students make when working with reference angles in the third quadrant is to take the angular distance to the y-axis rather than the distance from the x-axis (since this is the only one of the quadrants that requires going backwards rather than forwards to find a reference angle). While students did not encounter any problems with this today, I reminded them to be very careful about this and will pay close attention to their success or difficulties with this topic moving forward.

V. InTASC or COE Standard and Rationale - How does this lesson represent my progress toward gaining mastery of the skills of an effective teacher? This lesson is a good example of InTASC or COE ____ because . . . • InTASC 4 (Content Knowledge): This lesson was specifically designed to link the new concepts (trigonometric values of the three special acute angles) to students’ existing knowledge from Geometry (properties of 30-60-90 and 45-45-90 right triangles). In a way, this was a somewhat risky strategy since many of the students took Geometry over two years ago, making it uncertain how much knowledge they had actually retained, but my faith in this particular class paid off: all students were able to draw the connections between the two fields. In addition, Geometer’s Sketchpad is a technological resource not available in the school (CCPS does not have a system-wide software license); instead, I purchased a copy for my personal computer and used this for the demonstration. • COE 11 (Use of Technology): Although technology use has been shown to have significant benefits for students—and I certainly feel that was the case today—any integration must be more than simply using technology for technology’s sake. In this instance, I made sure that the Geometer’s Sketchpad demonstration would give students something that they could not get otherwise—in this case, an interactive, step-by-step demonstration of how the functions were derived. Not only would it have taken significantly longer to illustrate these derivations using freehand tools such as a compass, but it also would not have been possible to add touches such as shading of the triangle in question to ensure students were focusing on it and dynamic measurements of the relevant side lengths and angle measures. All steps were also clearly delineated with steps and text and explained verbally, making this lesson accessible for both auditory and visual learners.

30 Degrees 60 Degrees

45 Degrees

Introduction to Radians (March 5)

Linear and Angular Speed (March 12)