 # Standard #: MAFS.912.G-SRT.3.8

This document was generated on CPALMS - www.cpalms.org

Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

### General Information

Subject Area: Mathematics
Domain-Subdomain: Geometry: Similarity, Right Triangles, & Trigonometry
Cluster: Level 2: Basic Application of Skills & Concepts
Cluster: Define trigonometric ratios and solve problems involving right triangles. (Geometry - Major Cluster) -

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

Date of Last Rating: 02/14
Status: State Board Approved
Assessed: Yes

### Test Item Specifications

Also assesses:

MAFS.912.G-SRT.3.6

MAFS.912.G-SRT.3.7

Assessment Limits :
Items will assess only sine, cosine, and tangent to determine the
length of a side or an angle measure.
Calculator :

Neutral

Clarification :
Students will use trigonometric ratios and the Pythagorean theorem
to solve right triangles in applied problems.

Students will use similarity to explain the definition of trigonometric
ratios for acute angles.

Students will explain the relationship between sine and cosine of
complementary angles.

Students will use the relationship between sine and cosine of
complementary angles.

Stimulus Attributes :
For G-SRT.3.8, items must be set in a real-world context.

For G-SRT.3.6 and G-SRT.3.7, items must be set in a mathematical
context.

For G-SRT.3.8, items may require the student to apply the basic
modeling cycle.

Response Attributes :
Items may require the student to find equivalent ratios.

Items may require the student to use or choose the correct unit of
measure.

Multiple-choice options may be written as a trigonometric equation.

Equation Editor items may require the student to use the inverse
trigonometric function to write an expression.

### Sample Test Items (1)

 Test Item # Question Difficulty Type Sample Item 1 In the 1990s, engineers restored the building so that angle y changed from 5.5º to 3.99º.To the nearest hundredth of a meter, how much did the restoration change the height of the Leaning Tower of Pisa? N/A EE: Equation Editor

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#### Related Access Points

 Access Point Number Access Point Title MAFS.912.G-SRT.3.AP.8a Apply both trigonometric ratios and Pythagorean Theorem to solve application problems involving right triangles.

#### Assessments

 Name Description Sample 1 - High School Geometry State Interim Assessment This is a State Interim Assessment for 9th-12th grade. Sample 4 - High School Geometry State Interim Assessment This is a State Interim Assessment for 9th-12th grades. Sample 3 - High School Geometry State Interim Assessment This is a State Interim Assessment for 9th-12th grade. Sample 2 - High School Geometry State Interim Assessment This is a State Interim Assessment for 9th-12th grade.

#### Formative Assessments

 Name Description Washington Monument Students are asked to find the angle of elevation in a real world situation modeled by a right triangle. Step Up Students are asked to explain the relationship among angles in a diagram involving a right triangle and to find one angle of the right triangle. River Width Students are asked to find an unknown length in a real world context requiring right triangle trigonometry. Perilous Plunge Students are asked to find an unknown length in a real world context requiring right triangle trigonometry. Lighthouse Keeper Students are asked to find the difference between two lengths in a real world context requiring right triangle trigonometry. Holiday Lights Students are asked to solve a problem in a real world context requiring the use of the Pythagorean Theorem. Will It Fit? Students are asked to solve a problem in a real world context using the Pythagorean Theorem. TV Size Students are asked to solve a problem in a real world context requiring the use of the Pythagorean Theorem.

#### Lesson Study Resource Kit

 Name Description The Motion of Objects This 9-12 Lesson study resource kit is designed to engage teachers of physical science and physics in the planning and design of an instructional unit and research lesson pertaining to the motion of objects. Included in this resource kit are unit plans, concept progressions, formative and summative assessments, complex informational texts, and etc. that align to relevant NGSSS science, and the new Florida standards for mathematics and English language arts.

#### Original Student Tutorial

 Name Description Around the World with Right Triangles Learn how to use trigonometric ratios to solve a real-world application. There are many famous monuments across the world. The measurements of these monuments were often found using trigonometric ratios. Today, there are devices that use laser beams to measure distances and heights, but trigonometric ratios are still widely used.

#### Perspectives Video: Expert

 Name Description Oceanography & Math A discussion describing ocean currents studied by a physical oceanographer does and how math is involved.  Download the CPALMS Perspectives video student note taking guide.

#### Presentation/Slideshow

 Name Description The Pythagorean Theorem: Geometry’s Most Elegant Theorem This lesson teaches students about the history of the Pythagorean theorem, along with proofs and applications. It is geared toward high school Geometry students that have completed a year of Algebra and addresses the following national standards of the National Council of Teachers of Mathematics and the Mid-continent Research for Education and Learning: 1) Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships; 2) Use visualization, spatial reasoning, and geometric modeling to solve problems; 3) Understand and apply basic and advanced properties of the concepts of geometry; and 4) Use the Pythagorean theorem and its converse and properties of special right triangles to solve mathematical and real-world problems. The video portion is about thirty minutes, and with breaks could be completed in 50 minutes. (You may consider completing over two classes, particularly if you want to allow more time for activities or do some of the enrichment material). These activities could be done individually, in pairs, or groups. I think 2 or 3 students is optimal. The materials required for the activities include scissors, tape, string and markers.

#### Teaching Idea

 Name Description Measuring the Distance to Nearby Stars Using Parallax This video provides a very complete and detailed overview of the parallax effect and how it can be used to measure astronomical distances using the tangent function. A number of student activities are presented throughout the 26 minute video, so students can have the opportunity to engage in measuring distances to stars and other local landmarks and can try making the required calculations on their own.The relevance of this concept to other fields, such as surveying, is also noted in the video.

#### Tutorials

 Name Description Using Trigonometry to solve for missing information This tutorial will show students how to use trigonometry to solve for missing information in right triangles. This video shows worked examples using trigonometric ratios to solve for missing information and evaluate other trigonometric ratios. Basic Trigonometry This tutorial gives an introduction to trigonometry. This resource discusses the three basic trigonometry functions, sine, cosine, and tangent. Projectile at an angle This video discusses how to figure out the horizontal displacement for a projectile launched at an angle. LSSS Tutorial: Introduction to Vectors and Scalars This resource is intended to serve as a concise introduction to vector and scalar quantities for teachers of secondary math and science. It provides definitions of vectors and scalars as well as physical examples of each type of quantity, and also illustrates the differences between these two types of quantities in both one and two dimensions, through determinations of both distance (scalar) and displacement (vector).

#### Video/Audio/Animation

 Name Description MIT BLOSSOMS - The Juice Seller’s Problem "This video lesson presents a real world problem that can be solved by using the Pythagorean theorem. The problem faces a juice seller daily. He has equilateral barrels with equal heights and he always tries to empty the juice of two barrels into a third barrel that has a volume equal to the sum of the volumes of the two barrels. This juice seller wants to find a simple way to help him select the right barrel without wasting time, and without any calculations - since he is ignorant of mathematics. The prerequisite for this lesson includes knowledge of the following: the Pythagorean theorem; calculation of a triangle's area knowing the angle between its two sides; cosine rule; calculation of a circle's area; and calculation of the areas and volumes of solids with regular bases. Materials necessary include: equilateral containers of equal heights; sand; and measuring devices. Examples of in-class activities for the breaks between video segments include class discussions, individual calculations and small group problem solving." (from MIT Blossoms' "Pythagoras and the Juice Seller")

#### Virtual Manipulatives

 Name Description Right Triangle Solver This virtual manipulative will help the students in understanding that the relationships found in right triangles can be used to solve many applied problems in science and engineering. The right triangle solver manipulative displays a triangle with some its sides and angles given. The student is then asked to determine values of the remaining sides and angles by choosing a workable strategy. Triangle Solver The triangle solver manipulative displays a triangle with some of its sides and angles given. The students are then asked to determine values of the remaining sides and angles. Students are motivated to choose a workable strategy such as using the Pythagorean theorem, the sine, cosine, tangent relationships, the law of sines, or the law of cosines. They are directed through the key steps of the chosen strategy to find the unknown sides and the angles. Pythagorean Theorem Manipulatives This web address, from the National Library of Virtual Manipulatives, will help teachers and students validate the Pythagorean Theorem both geometrically and algebraically. It can be used interactively with the Smartboard and the Promethean Board to create a better understanding of the topic. Demonstrate the Pythagorean Theorem Representation to illustrate the Pythagorean Theorem.

#### Original Student Tutorial

 Name Description Around the World with Right Triangles: Learn how to use trigonometric ratios to solve a real-world application. There are many famous monuments across the world. The measurements of these monuments were often found using trigonometric ratios. Today, there are devices that use laser beams to measure distances and heights, but trigonometric ratios are still widely used.

#### Presentation/Slideshow

 Name Description The Pythagorean Theorem: Geometry’s Most Elegant Theorem: This lesson teaches students about the history of the Pythagorean theorem, along with proofs and applications. It is geared toward high school Geometry students that have completed a year of Algebra and addresses the following national standards of the National Council of Teachers of Mathematics and the Mid-continent Research for Education and Learning: 1) Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships; 2) Use visualization, spatial reasoning, and geometric modeling to solve problems; 3) Understand and apply basic and advanced properties of the concepts of geometry; and 4) Use the Pythagorean theorem and its converse and properties of special right triangles to solve mathematical and real-world problems. The video portion is about thirty minutes, and with breaks could be completed in 50 minutes. (You may consider completing over two classes, particularly if you want to allow more time for activities or do some of the enrichment material). These activities could be done individually, in pairs, or groups. I think 2 or 3 students is optimal. The materials required for the activities include scissors, tape, string and markers.

#### Tutorials

 Name Description Using Trigonometry to solve for missing information: This tutorial will show students how to use trigonometry to solve for missing information in right triangles. This video shows worked examples using trigonometric ratios to solve for missing information and evaluate other trigonometric ratios. Basic Trigonometry: This tutorial gives an introduction to trigonometry. This resource discusses the three basic trigonometry functions, sine, cosine, and tangent. Projectile at an angle: This video discusses how to figure out the horizontal displacement for a projectile launched at an angle.

#### Virtual Manipulatives

 Name Description Right Triangle Solver: This virtual manipulative will help the students in understanding that the relationships found in right triangles can be used to solve many applied problems in science and engineering. The right triangle solver manipulative displays a triangle with some its sides and angles given. The student is then asked to determine values of the remaining sides and angles by choosing a workable strategy. Triangle Solver: The triangle solver manipulative displays a triangle with some of its sides and angles given. The students are then asked to determine values of the remaining sides and angles. Students are motivated to choose a workable strategy such as using the Pythagorean theorem, the sine, cosine, tangent relationships, the law of sines, or the law of cosines. They are directed through the key steps of the chosen strategy to find the unknown sides and the angles. Pythagorean Theorem Manipulatives: This web address, from the National Library of Virtual Manipulatives, will help teachers and students validate the Pythagorean Theorem both geometrically and algebraically. It can be used interactively with the Smartboard and the Promethean Board to create a better understanding of the topic.