Why Is It Special?

Subject Area

Mathematics

Grade(s)

9, 10, 11, 12

Overview

In this Desmos activity, students will explore the special relationships between the sides of a 30-60-90 triangle and a 45-45-90 triangle. This is intended to be a before activity for a lesson on the relationships of special right triangles and their connection to trigonometry.

This activity results from the ALEX Resource Development Summit.

Phase

Before/Engage

Mathematics (2019) Grade(s): 09-12 - Geometry with Data Analysis

MA19.GDA.35
Discover and apply relationships in similar right triangles.

Vocabulary

Side ratios
Trigonometric ratios
Sine
Cosine
Tangent
Secant
Cosecant
Cotangent
Complementary anglesconverse

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.

Learning Objectives

The student will be able to find the sides of a special right triangle, either a 30-60-90 triangle or a 45-45-90 triangle using the special relationships between the sides.

Activity Details

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

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

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.

Variation Tips

Students could 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 on paper and check for mistakes.

Background / 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.

Owner (Author)

mrswhite

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