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slope. This unit builds on this extensive experience and grounds trigonometric ratios in familiar contexts.
The Rrst few lessons examine some special cases of similar right triangles to solidify the idea that any right triangles with a single congruent acute angle are similar. Two of these three lessons are optional. While the standards do not speciRcally call for special right triangles they are an opportunity to practice, build on important ideas, and are frequently included on college entrance exams. From there students generate data for the ratios of many sets of right triangles. This data is organized into a table which students apply to problems in the next several lessons. Once students have practice estimating both side lengths and angle measures using the table, they learn the names cosine, sine, and tangent.
After students practice looking up the sine, cosine, or tangent of a given angle in a calculator, they apply them to several contexts. Then they investigate the relationship between cosine and sine, proving the two ratios are equal for complementary angles. Next, students learn how to Rnd the acute angles in a right triangle given the side measures by using arcsin, arccos, and arctan. Students then apply all of these skills to a few scenarios.
G5 Solid Geometry
In previous grades, students have solved problems involving area, surface area, and volume for various solids. In grade 6, students worked with areas of triangles and quadrilaterals, as well as volumes of right rectangular prisms, including those with fractional edge lengths. In grade 7, students worked with areas of circles and solved problems involving the volume and surface area of various right prisms. In grade 8, students solved problems involving volumes of spheres, cones, and cylinders from given volume formulas.
In this unit, students practice spacial visualization in three-dimensions, study the eUect of dilation on area and volume, derive volume formulas using dissection arguments and Cavalieri's Principle, and apply volume formulas to solve problems involving surface area to volume ratios, density, cube roots, and square roots.
Students practice spacial visualization by examining cross sections and volumes of rotation. Students make cross sections of a pyramid by dilating a two-dimensional shape using a center of dilation that is not in the same plane as the shape. Students explore various volumes of rotation by rotating paper Rgures using a pencil as an axis of rotation.
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