ALEX Learning Activity

  

Why is it Special? Verifying Relationships of Right Triangles

A Learning Activity is a strategy a teacher chooses to actively engage students in learning a concept or skill using a digital tool/resource.

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  This learning activity provided by:  
Author: Lorie White
System:Lauderdale County
School:Rogers High School
  General Activity Information  
Activity ID: 2597
Title:
Why is it Special? Verifying Relationships of Right Triangles
Digital Tool/Resource:
What Makes It Special - Verifying Relationships
Web Address – URL:
Overview:

In this Desmos activity, students are guided through a series of questions to verify the relationships between the sides and angles of a 30°-60°-90° right triangle and a 45°-45°-90° triangle. This is intended to be a during activity for a lesson on discovering and applying relationships in similar right triangles. 

This activity results from the ALEX Resource Development Summit.
  Associated Standards and Objectives  
Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
35. Discover and apply relationships in similar right triangles.

a. Derive and apply the constant ratios of the sides in special right triangles (45o-45o-90o and 30o-60o-90o).

b. Use similarity to explore and define basic trigonometric ratios, including sine ratio, cosine ratio, and tangent ratio.

c. Explain and use the relationship between the sine and cosine of complementary angles.

d. Demonstrate the converse of the Pythagorean Theorem.

e. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems, including finding areas of regular polygons.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Given a collection of right triangles, discover and apply relationships in similar right triangles.
  • Derive and apply the ratios of the sides of the original triangles to the ratios of the sides of the similar triangles.
  • Communicate observations made about changes (or no change) to such ratios as the length of the side opposite an angle to the hypotenuse, or the side opposite the angle to the side adjacent, as the size of the angle changes or in the case of similar triangles, remains the same.
    Summarize these observations by defining the six trigonometric ratios.
  • Explain why the two smallest angles must be complements.
  • Compare the side ratios of opposite/hypotenuse and adjacent/hypotenuse for each of these angles and discuss conclusions.


Given a contextual situation involving right triangles,
  • Create a drawing to model the situation.
  • Find the missing sides and/or angles using trigonometric ratios.
  • Find the missing sides using the Pythagorean Theorem.
  • Use the above information to interpret results in the context of the situation, including finding the areas of regular polygons.
Teacher Vocabulary:
  • Side ratios
  • Trigonometric ratios
  • Sine
  • Cosine
  • Tangent
  • Secant
  • Cosecant
  • Cotangent
  • Complementary angles converse
Knowledge:
Students know:
  • Techniques to construct similar triangles.
  • Properties of similar triangles.
  • Methods for finding sine and cosine ratios in a right triangle (e.g., use of triangle properties: similarity. Pythagorean Theorem. isosceles and equilateral characteristics for 45-45-90 and 30-60-90 triangles and technology for others).
  • Methods of using the trigonometric ratios to solve for sides or angles in a right triangle.
  • The Pythagorean Theorem and its use in solving for unknown parts of a right triangle.
Skills:
Students are able to:
  • Accurately find the side ratios of triangles.
  • Explain and justify relationships between the side ratios of a right triangle and the angles of a right triangle.
Understanding:
Students understand that:
  • The ratios of the sides of right triangles are dependent on the size of the angles of the triangle.
  • The sine of an angle is equal to the cosine of the complement of the angle.
  • Switching between using a given angle or its complement and between sine or cosine ratios may be used when solving contextual problems.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.35.1: Define trigonometric (sine, cosine and tangent) ratios for acute angles, complementary angles, and Pythagorean Theorem.
GEO.35.2: Simplify, multiply, and divide radicals.
GEO.35.3:Discuss the relationship between sine and cosine angles within a triangle.
GEO.35.4: Solve equations using trigonometric ratios.
GEO.35.5: Apply properties of similarity to demonstrate the trigonometric ratios of right triangles.
GEO.35.6: Use Pythagorean Theorem to find the missing side of a right triangle.
GEO.35.7: Create an equation using the given information of a right triangle.
GEO.35.8: Identify the parts of a right triangle.
Examples: legs, hypotenuse, right angle.

Prior Knowledge Skills:
  • Demonstrate how to find square roots.
  • Identify right triangles.
  • Solve problems using the Pythagorean Theorem.
  • Recognize ordered pairs (x, y).
  • Recall how to name points on a Cartesian plane using ordered pairs.
  • Identify right triangles.
  • Solve problems using the Pythagorean Theorem.
  • Discuss strategies for solving real-world and mathematical problems.
  • Recognize examples of right triangles.
  • Define a right angle, Pythagorean Theorem, converse, and proof.
  • Apply properties to find missing angle measures.
  • Identify a transversal.
  • Identify exterior, interior, vertical, adjacent, alternate interior, alternate exterior, and corresponding angles.
  • Identify attributes of triangles.
  • Define exterior, interior, vertical, adjacent, alternate interior, alternate exterior, corresponding, and transversal angles.
  • Identify right and straight angles.
  • Identify all types of angles.
  • Identify proportional relationships.
  • Locate/use scale on a map.
  • Define scale, scale drawings, length, area, and geometric figures.
  • Recall how to find unit rates using ratios.
  • Define unit rate, proportions, area, length, and ratio.
  • Analyze the area of other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes.
  • Define area, special quadrilaterals, right triangles, and polygons.
  • Recognize and demonstrate that two right triangles make a rectangle.
  • Select manipulatives to demonstrate how to compose and decompose triangles and other shapes.
  • Explain how to find the area for rectangles.
  • Demonstrate how the area of a rectangle is equal to the sum of the area of two equal right triangles.
  • Apply area formulas to solve real-world mathematical problems.
  • Recognize polygons.
  • Restate real-world problems or mathematical problems.
  • Calculate unit rate or rate by using ratios or proportions.
  • Create a ratio or proportion from a given word problem, diagram, table, or equation.
  • Define ratio, rate, proportion, percent, equivalent, input, output, ordered pairs, diagram, unit rate, and table.
  • Form a ratio.
  • Solve a proportion using part over whole equals percent over 100.
  • Identify a proportion from given information.
  • Calculate a proportion for missing information.
  • Create a proportion or ratio from a given word problem.
Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.36 Use geometric shapes to describe real-world objects.
Learning Objectives:

Students will be able to use the Pythagorean Theorem to solve find the relationships between the sides and angles in a 30°-60°-90° right triangle and a 45°-45°-90° triangle.  
  Strategies, Preparations and Variations  
Phase:
During/Explore/Explain
Activity:

  1. Direct each student to the Desmos learning activity, What Makes It Special? Verifying Relationships.

  2. Instruct the students to follow and answer the prompts provided in the activity independently.  

  3. Remind the students that they have previously studied the Pythagorean Theorem, which will help find the relationships between the sides.

  4. The teacher should monitor students' progress on Desmos to identify problems or gaps in learning.
Assessment Strategies:

Once the students share their answers with the class, they can see three other students' answers for a comparison of the correctness of their work. The teacher can assess the students' answers in Desmos as well by checking the students' responses. The teacher should correct the students' mistakes or misunderstandings.
Advanced Preparation:

The teacher will need to ensure the classroom has the following:

1) internet connection (WiFi).

2) classroom set of computers, iPads, or smart devices. (If allowed students can use their own personal devices.)

Prior to the lesson, the teacher should be familiar with Desmos and should create a free class code for the activity. If the teacher does not have a Desmos account, he/she will need to create a free Desmos account. Prior to the lesson, the teacher may want to review the Pythagorean Theorem and altitude.
Variation Tips (optional):

Students could work in pairs to complete this activity. If students do not have access to a device, the activity could be done as a whole group activity. If using with a whole group, the teacher should have students write their answers on whiteboards or paper and check for mistakes. 
Notes or Recommendations (optional):

ALCOS 2019

35. Discover and apply relationships in similar right triangles. 
a. Derive and apply the constant ratios of the sides in special right triangles (45˚-45˚-90˚ and 30˚-60˚-90˚).
b. Use similarity to explore and define basic trigonometric ratios, including sine ratio, cosine ratio, and tangent ratio.
c. Explain and use the relationship between the sine and cosine of complementary angles. 
d. Demonstrate the converse of the Pythagorean Theorem.
e. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems, including finding areas of regular polygons.
  Keywords and Search Tags  
Keywords and Search Tags: Pythagorean Theorem, Right Angle Trigonometry, Special Right Triangle Relationship