Investigation and Justification (Proof) Thread

Transcription

1 Concept Category 3 (CC3): Triangle Trigonometry Grounded in students study of similar triangles in CC2, students consider slope triangles in CC3 to learn about the relationship between the angles and the side lengths of a right triangle developing the tangent ratio. Finish CC3a by revisiting similar slope triangles, this time focusing on relationships between the legs and the hypotenuse of these right triangles in order to develop the sine and cosine ratios. Students continue to model real life situations using right triangles and to solve problems using all three trigonometric ratios. They are also introduced to inverse trigonometric ratios and learn to use side ratios to find missing angles in right triangles. CC3b extends what students know about right triangles to include special cases such as and triangles and those with side lengths that are Pythagorean Triples. This section has strong connections to work from CC1 & CC2, as students utilize the reflection symmetry of both equilateral triangles and squares to unlock the ratios of and triangles. Then focus on completing a toolkit for finding missing parts of non-right triangles. Begin by asking students to identify the types of information needed to find all of the missing sides and angles of a triangle. Through this exercise, students also identify triangles for which they do not yet have the tools to find missing parts. Students notice that they do have enough tools to find the angles and side lengths of right triangles. This leads to the question, What if the triangle is not a right triangle? Students then develop the Law of Sines and Law of Cosines so that they have a complete set of tools to find the other missing parts of any triangle (when sufficient information is provided). The CC concludes with students looking at different application problems using triangles and identifying which tools are most useful in each situation. In addition, investigate the ambiguous case of triangles: SSA. This topic is offered for accelerated classes or those that could benefit from a complete view of the relationships between the sides and angles of a triangle. By the end of CC3, your students will have six triangle tools : the Pythagorean Theorem (from CC1), tangent, sine, cosine, Law of Sines, and Law of Cosines. In addition, students will also know triangle shortcuts (for and triangles) and know how to use inverse trigonometric functions in order to solve for an angle of a right triangle. Algebra Thread The equation writing and solving skills in this CC are extended as students develop and apply the trigonometric ratio tangent. Students interpret real life situations, create diagrams to model those situations, and then use the diagrams to write equations. Algebra review of other topics, including applications involving area and perimeter, carries through this CC. In addition, students review how to solve quadratic equations in assignments in preparation for solving geometric problems in which quadratic equations arise. It is assumed that students have some familiarity with solving quadratic equations. While factoring and using the Zero Product Property are viable and useful strategies to solve simple quadratics, this CC will not focus on factoring. Instead, it provides the Quadratic Formula as a tool to solve quadratic equations. The development of the Law of Sines and the Law of Cosines will require students to manipulate equations with multiple variables, substitute equivalent expressions, and simplify complex equations. Some students may feel uneasy working with equations that contain four or more variables or constants, and they will need to be reminded that the simplification steps that they use comfortably with numbers are still applicable to these equations. Investigation and Justification (Proof) Thread

2 Students continue to develop their knowledge of how to investigate and draw reasoned conclusions through this CC. Students consider the validity of given conjectures in CC3 and create their own conjectures. While they are not yet writing proofs or formal arguments to support their new conjectures, students should be justifying their thinking verbally with their peers and writing down explanations of their conclusions. Where is this going? This CC continues to build on the question, What can I figure out about a triangle? that was introduced in CC1 with the Pythagorean Theorem. Trigonometry is used throughout the rest of the CCs as students solve complex problems involving area, perimeter and surface area, and as students develop an algorithm for finding the area of regular polygons. By the end of this CC, students will have a complete set of tools to find missing sides and angles of triangles when sufficient information is given and the triangle is not ambiguous. In later CCs, students will use these tools as they investigate properties of regular polygons, calculate volume and surface area of prisms and pyramids, and solve different application problems. Create a diagram based on information in a word problem, and identify a right triangle within that diagram, as in problems 4-43, 4-50, 4-83, 4-113, and CL In right triangles where one leg and an angle are given, use a calculator to find missing sides using the tangent ratio, as in the problems in the above bullet, and as in problems 4-39, 4-63, 4-74, CL 4-122, and CL Find the missing sides or angles of right triangles using sine, cosine, tangent, as in problems 5-17, 5-18, 5-44, 5-46, 5-100, 5-137, CL 5-139(a-d), and CL 5-143(a), and their inverse functions, as in problems 5-30, 5-77, 5-103, and CL 5-139(e-g). Problems may be presented as triangle diagrams in different orientations, or word problems with a diagram provided or instructions to draw one. Students can be asked to find the area and/or perimeter of a shape where they will need to use their knowledge of trigonometry to find missing lengths, as in problems 5-11, 5-33, 5-52, 5-129, and Find the slope angle or the slope of the line segment on a coordinate grid using the tangent ratio, as in problems 5-42, 5-102, 5-113, and CL First semester ends here, CC3 will pick up Second Semester with the rest of CC3 & continue on. Recognize and apply the side ratios in triangles and triangles, as in problems 5-52, 5-53(d), 5-64(b) and (d), 5-91, 5-127, 5-135, and CL Problems may present triangles independently or joined as part of a larger shape. Recognize the smallest Pythagorean Triples and use them as shortcuts to find missing sides in other triangles, as in problems 5-64(a) and (c), 5-129, parts of 5-138, and CL (a) and (b). Understand that it is possible to find missing sides on non-right triangles, as in problems 5-90, 5-111, 5-126, 5-134, and CL It is acceptable for students to create right triangles within a diagram in order to find missing parts. Law of Sines and Law of Cosines. However, students can find missing sides on non-right triangles by making right triangles within a diagram. Sample Problems referred to above: Create a diagram based on information in a word problem, and identify a right triangle within that diagram, as in problems 4-43, 4-50, 4-83, 4-113, and CL

3 In right triangles where one leg and an angle are given, use a calculator to find missing sides using the tangent ratio, as in the problems in the above bullet, and as in problems 4-39, 4-63, 4-74, CL 4-122, and CL

4 Find the missing sides or angles of right triangles using sine, cosine, tangent, as in problems 5-17, 5-18, 5-44, 5-46, 5-100, 5-137, CL 5-139(a-d), and CL 5-143(a), and their inverse functions, as in problems 5-30, 5-77, 5-103, and CL 5-139(e-g). Problems may be presented as triangle diagrams in different orientations, or word problems with a diagram provided or instructions to draw one.

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7 Students can be asked to find the area and/or perimeter of a shape where they will need to use their knowledge of trigonometry to find missing lengths, as in problems 5-11, 5-33, 5-52, 5-129, and

8 Find the slope angle or the slope of the line segment on a coordinate grid using the tangent ratio, as in problems 5-42, 5-102, 5-113, and CL Recognize and apply the side ratios in triangles and triangles, as in problems 5-52, 5-53(d), 5-64(b) and (d), 5-91, 5-127, 5-135, and CL Problems may present triangles independently or joined as part of a larger shape.

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10 Recognize the smallest Pythagorean Triples and use them as shortcuts to find missing sides in other triangles, as in problems 5-64(a) and (c), 5-129, parts of 5-138, and CL (a) and (b).

11 Understand that it is possible to find missing sides on non-right triangles, as in problems 5-90, 5-111, 5-126, 5-134, and CL It is acceptable for students to create right triangles within a diagram in order to find missing parts.

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