Skills, Objectives, Materials, and Ideas

• Geometry and Algebra II
• Knowledge of the six trigonometric functions (and the ability to convert trig functions into sine and cosine)
• Students should know the difference between solving equations and informally proving/verifying trig identities
• Knowledge of basic Fundamental Trig Identities (reciprocals, quotients, Pythagorean, odd-even relationships)
• Knowledge of advanced Algebra techniques for simplifying complex expressions (including factoring, simplifying rational expressions and simplifying complex rational expressions)
• Knowledge of Mathematics Properties including Commutative Property of Addition and Multiplication, Associative Property of Addition and Multiplication, Distributive Property, Identity Property of Addition and Multiplication, Additive Inverse Property, etc.

• Students will verify (and informally prove) fundamental trigonometric identities utilizing reciprocal, quotient, Pythagorean and odd-even relationships.
• Students will arrange slips of overhead paper into step-by-step iterations to illustrate the proper procedure for informally proving fundamental trigonometric identities.
• Students will also match slips of transparency paper containing proper Algebraic properties and fundamental identities as explanations for all iterations of the trig verification process.
• Students will present their informal proofs/verifications in front of the class utilizing an overhead projector to display the overhead strips of paper in the proper order accompanied by the accurate explanations.
• Students will be able to rearrange strips of overhead paper into proper verifications (with explanations) even if some of the strips contain extraneous information or if extra strips actually provide the steps to prove the identity using another procedure or if the extra strips could be used to prove the other side of the identity.
• Students will collaborate with other groups of students to observe alternate strategies for informally proving/ verifying trigonometric identities.

• Prepared kits containing overhead strips of transparency paper that form single or multiple methods for informal proofs/verifications of given identities (teacher samples/templates are included in this lab)
• Scissors (for preparation)
• Freezer quart bags with write-on labels (e.g. Ziploc Freezer Quart Bags)
• Student template for re-arranging strips of overhead paper at their desks
• Overhead transparency template for student displays
• Overhead projector for presentations (focused and arranged at a proper distance so that all students can see the detailed iterations of the verification process clearly throughout the classroom)

This activity is an excellent introduction to verifying identities because the steps in the kit provide intricate details and explanations that students tend to ignore when left to their own devices. Used as an introduction to the trig identity verification process, this activity will inspire students to provide more detailed steps and explanations when they are forced to informally write the steps to these identities on their own accord after the lab experience has concluded. Additionally, the lab contains three challenges/rounds: packets that only contain slips for proving one side in one manner, packets that include slips for proving both possible sides in only one manner for each side and a final packet that contains slips for proving both sides in multiple manners. The final packet should be used as the ultimate challenge and to see if different lab pairs use different slips to prove the same problem different ways. The final packet will also show students that there will be many possible ways to verify some of the identities that they will encounter.

Enrichment/Alternate Lab Ideas:

Additionally, this activity could be used in a similar manner to verify Geometric proofs (e.g. SAS, SSS, etc.) or to verify other trig identities not included in this lab (e.g. sum/difference formulas, half angle formulas, double angle formulas, etc.). Simply utilize the template (or change it to meet specialized objectives) to create innovative packets with different transparency slips.

Note: Since some of the packets contain multiple ways to verify identities, there may be duplicate explanations/work iterations on various slips of transparency paper.
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