Students develop an understanding of right triangles through an introduction to trigonometry, building an appreciation for the similarity of triangles as the basis for developing the Pythagorean theorem.
In Unit 4, Right Triangles & Trigonometry, students develop a deep understanding of right triangles through an introduction to trigonometry and the Pythagorean theorem. Students build an appreciation for how similarity of triangles is the basis for developing the Pythagorean theorem and trigonometric properties.
This unit begins with Topic A, Right Triangle Properties and Side-Length Relationships. Students define angle and side-length relationships in right triangles. Students use similarity to prove the Pythagorean theorem and the converse of the Pythagorean theorem. Students develop the algebraic tools to perform operations with radicals. In Topic B, Right Triangle Trigonometry, and Topic C, Applications of Right Triangle Trigonometry, students define trigonometric ratios and make connections to the Pythagorean theorem. Students gain practice with determining an appropriate strategy for solving right triangles. This skill is extended in Topic D, the Unit Circle, where students are introduced to the unit circle and reference angles. From here, students describe how non-right triangles can be solved using the Law of Sines and Law of Cosines, in Topic E. These skills are critical for students' ability to understand calculus and integrals in future years.
There are several lessons in this unit that do not have an explicit common core standard alignment. We have identified that these are important concepts to be introduced in geometry in order for students to access Algebra II and AP Calculus.
Pacing: 21 instructional days (19 lessons, 1 flex day, 1 assessment day)
This assessment accompanies Unit 4 and should be given on the suggested assessment day or after completing the unit.
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Internalization of Standards via the Unit Assessment
Internalization of Trajectory of Unit
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Altitude | Leg | Hypotenuse | Opposite | Adjacent |
Pythagorean theorem | Converse of the Pythagorean theorem | Radical | Radicand | Rationalize the denominator |
Sine | Cosine | Tangent | Trigonometric ratio | Arcsine ($${\mathrm{sin}^{-1}(\theta)}$$) |
Arccosine ($${\mathrm{cos}^{-1}(\theta)}$$) | Arctangent ($${\mathrm{tan}^{-1}(\theta)}$$) | Angle of elevation | Angle of depression | Standard position |
Unit circle | Reference angle | Area formula for non-right triangles | Law of Sines | Law of Cosines |
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Key: Major Cluster Supporting Cluster Additional Cluster
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