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Properties of Right Triangles: Is This the Right Angle?

 

This is one lesson in a larger Unit Plan that introduces triangular relationships through the lens of building a treehouse with a variety of triangular characteristics.  It has proofs, real world problems, constructions, a unit project, assessments, and more. Lesson 15, “Is This a Right Angle?", is a hands-on exercies that includes a writing assignment which requires students to use the Pythagorean Theorem and special right triangles.  Resource includes a grading rubric.  The entire unit plan can be found here. This is a portion of a Unit Project, however, it could be used as a stand alone assignment as well. The students have to design braces and walls based on special right triangles. The lesson includes a suggestion to help with differential instruction.

Weblink address (URL): 
Properties of Right Triangles: Is This the Right Angle?

Standards & Objectives

Learning objectives: 
  • Objectives:
  • represent geometric figures, such as points, lines, planes, segments, rays, and angles pictorially with proper identification and distinguish between undefined and defined terms.
  • validate conclusions by constructing logical arguments using both formal and informal methods with direct and indirect reasoning.
  • construct formal and informal proofs by applying definitions, theorems, and postulates related to such topics as
  • complementary, supplementary, vertical angles, angles formed by perpendicular lines, and justify the steps.
  • make conjectures and justify congruence relationships with an emphasis on triangles and employ these relationships to solve problems.
  • investigate measures of angles and lengths of segments to determine the existence of a triangle (triangle inequality) and to establish the relationship between the measures of the angles and the length of the sides (with and without technology).
  • verify and justify the basis for the trigonometric ratios by applying properties of similar triangles  and use the results to find inaccessible heights and distances.  Using the ratios of similar triangles to find unknown side lengths and angle measures, construct a physical model that illustrates the use of a scale drawing in a real-world situation.
  • apply the Pythagorean Theorem and its converse to solve real-world problems and derive the special right triangle relationships (i.e. 30-60-90, 45-45-90).
  • find angle measures of interior and exterior angles; given a polygon, find the length of sides from given data; and use properties of regular polygons to find any unknown measurements of sides or angles.
  • derive and justify formulas for area, perimeter, surface area, and volume using nets and apply them to solve real-world problems.
  • construct a triangle’s medians, altitudes, angle and perpendicular bisectors using various methods; and develop logical concepts about their relationships to be used in solving real-world problems.
  • identify a real life situation that involves similarity in two or three dimensions; pose a question; make a hypothesis as to the answer, develop, justify, and implement a method to collect, organize, and analyze related data; generalize the results to make a conclusion; compare the hypothesis and the conclusion; present the project numerically, analytically, graphically and verbally using the predictive and analytic tools of algebra and geometry (with and without technology).
Essential and guiding questions: 
  • How can 21st Century tools improvee the study of triangles?
  • How can triangular relationships affect your everday life?
  • Why study triangular relationships?
  • How do the properties of triangles influence their usage?
  • What are the limits of geometrical representation and moddeling?

Helpful Hints

Materials: 

  • Sticky notes
  • markers
  • poster paper
  • activity sheets
  • journals
  • straightedge
  • protractors
  • spaghetti or Wikki Stix 
  • plain white paper
  • drinking straws (dry spaghetti, etc.)
  • rulers
  •  calculators
  • computers with internet access
  • compasses 
  • scissors 
  • cardboard
  • poster board 
  • patty paper
  • Miras 
  • tape
  • SketchPad software (optional)

References

Contributors: 
Lisa Leach
Originator: 
Ayers Institute for Learning & Innovation