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VITAL INFORMATION

Subject(s):

Mathematics

Topic or Unit of Study:

Trigonometry

Grade/Level:

Objective:

• Students will use the Pythagorean Theorem to discover the trigonometric relationships and identities.
• Students will build upon their knowledge of the unit circle and the relationship of right triangles with sides x, y, and r to trigonometry.

Summary:

Through hands-on construction, students will form and share with their classmates trigonometric relationships and identities.

CLASS SIZE: 24

IMPLEMENTATION

Learning Context:

• This lesson is the basis for understanding how to manipulate the trigonometric equations in order to succeed in solving them for specific numbers.
• The students will utilize their knowledge of the unit circle and Pythagoras.

Procedure:

• Launch: As we begin class, students will copy the following definition from the board:
"An equation involving the trigonometric functions which is valid for all values for the angle for which the functions are defined is called a trigonometric identity. A trigonometric identity is verified by transforming one member (your choice) into the other. In general, one begins with the more complicated side. In some cases each side is transformed into the same new form." As the students copy the definition from the board, discuss the class by telling them that they will be using the index cards on their tables, in groups of two, to form new trigonometric relationships and identities. Pair the students according to students' levels- have more advanced learners with more struggling students. (10 min)

• Investigation:
The index cards are labeled x, y or r and there are four of each. I will begin the investigation using a transparency on an overhead. My "cards" are transparencies. I show how if I form (y/r) it relates to sin(x). Students will know this from their work with the unit circle. Then it follows that (x/r) equals cos(x). Because the unit circle uses right triangles to form values, I apply the pythagorean theorem of a^2 + b^2 = c^2 to x, y, and r (where r=1 on a unit circle). Ask the students for suggestions on how to manipulate my cards to reflect the Pythagorean Theorem. This will eventually lead to the formulation of (y/r)^2 + (x/r)^2 = 1, or (sinx)^2 + (cosx)^2 = 1. This is the first relationship they will need to know in order to form more relationships and identities. The students will record this in their notes and then begin their own discovery. (35 min)

• Summary: The lesson will close with the students’ presentation of three identities/relationships they have discovered. Students will be asked to prove the identities they present on the overhead. (15 min)

LITERACY ASPECT: Students will be required to write down their findings as well as their peers' in their notebooks, exercising their use of mathematical language. They will also be responsible for discussing, in mathematical terms, their discoveries. They will also be required to write a daily reflection in their math journals.

FOLLOW-UP ACTIVITY:
Students will solve various identities on a handout or from their textbooks for homework.

Differentiated Instruction:

Sample Student Products:

Collaboration:

Students will work collaboratively & individually. Students will work in groups of 2.

Time Allotment:

1 class period. 1 Hr per class.

Author's Comments & Reflections:

Another teacher will need a resource such as a textbook that will allow him/her to implement this activity and successfully answer students' questions.

MATERIALS AND RESOURCES

Instructional Materials:

Resources:

Materials and resources:
• Index Cards
• Pencils, Markers
• Overhead
• Transparencies

STANDARDS & ASSESSMENT

Standards:

CA- California K-12 Academic Content Standards

• Subject : Mathematics

• Grade : Grades Eight Through Twelve
The standards for grades eight through twelve are organized differently from those for kindergarten through grade seven. In this section strands are not used for organizational purposes as they are in the elementary grades because the mathematics studied in grades eight through twelve falls naturally under disci-pline headings: algebra, geometry, and so forth. Many schools teach this material in traditional courses; others teach it in an integrated fashion. To allow local educational agencies and teachers flexibility in teaching the material, the stan-dards for grades eight through twelve do not mandate that a particular discipline be initiated and completed in a single grade. The core content of these subjects must be covered; students are expected to achieve the standards however these subjects are sequenced.

Standards are provided for algebra I, geometry, algebra II, trigonometry, math-ematical analysis, linear algebra, probability and statistics, Advanced Placement probability and statistics, and calculus. Many of the more advanced subjects are not taught in every middle school or high school. Moreover, schools and districts have different ways of combining the subject matter in these various disciplines. For example, many schools combine some trigonometry, mathematical analysis, and linear algebra to form a precalculus course. Some districts prefer offering trigonometry content with algebra II.

• Area : Trigonometry
Trigonometry uses the techniques that students have previously learned from the study of algebra and geometry. The trigonometric functions studied are defined geo-metrically rather than in terms of algebraic equations. Facility with these functions as well as the ability to prove basic identities regarding them is especially important for students intending to study calculus, more advanced mathematics, physics and other sciences, and engineering in college.

• Sub-Strand 3.0: Students know the identity cos 2 (x) + sin 2 (x) = 1:

Standard 3.1: Students prove that this identity is equivalent to the Pythagorean theorem (i.e., students can prove this identity by using the Pythagorean theorem and, con-versely, they can prove the Pythagorean theorem as a consequence of this identity).

Standard 3.2: Students prove other trigonometric identities and simplify others by using the identity cos 2 (x) + sin 2 (x) = 1. For example, students use this identity to prove that sec 2 (x)Ê =Ê tan 2 (x) + 1.

USA- NCTM (Nat. Council of Teachers of Mathematics): Principles & Standards for School Mathematics

• Area : Standards

• Level : Grades 9–12

• Topic : Algebra

• Instructional program descriptor : Represent and analyze mathematical situations and structures using algebraic symbols

Expectation : understand the meaning of equivalent forms of expressions, equations, inequalities, and relations;

Expectation : write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency—mentally or with paper and pencil in simple cases and using technology in all cases;

• Topic : Problem Solving

Instructional program descriptor : build new mathematical knowledge through problem solving;

Instructional program descriptor : solve problems that arise in mathematics and in other contexts;

Instructional program descriptor : apply and adapt a variety of appropriate strategies to solve problems;

Instructional program descriptor : monitor and reflect on the process of mathematical problem solving.

• Topic : Reasoning and Proof

Instructional program descriptor : recognize reasoning and proof as fundamental aspects of mathematics;

Instructional program descriptor : make and investigate mathematical conjectures;

Instructional program descriptor : develop and evaluate mathematical arguments and proofs;

Instructional program descriptor : select and use various types of reasoning and methods of proof.

• Topic : Communication

Instructional program descriptor : organize and consolidate their mathematical thinking through communication;

Instructional program descriptor : communicate their mathematical thinking coherently and clearly to peers, teachers, and others;

Instructional program descriptor : analyze and evaluate the mathematical thinking and strategies of others;

Instructional program descriptor : use the language of mathematics to express mathematical ideas precisely.

• Topic : Connections

Instructional program descriptor : recognize and use connections among mathematical ideas;

Instructional program descriptor : understand how mathematical ideas interconnect and build on one another to produce a coherent whole;

• Topic : Representation

Instructional program descriptor : create and use representations to organize, record, and communicate mathematical ideas;

Instructional program descriptor : select, apply, and translate among mathematical representations to solve problems;

Assessment/Rubrics:

• Students will be assessed during groupwork. Monitor the progression of the students by how much they are participating in active discovery of formulas and questions they are asked.
• Students will also be assessed through the presentations of the relationships and identities. Students should give clear proof of why the identity and/or relationship works with use of the overhead cards and be able to answer other students' questions about formulation.

Trigonometric Relationships and Identities