 Secondary Mathematics II

Strand: GEOMETRY - Similarity, Right Triangles, and Trigonometry (G.SRT)

Understand similarity in terms of similarity transformations (Standards G.SRT.1�3). Prove theorems involving similarity (Standards G.SRT.4�5). Define trigonometric ratios and solve problems involving right triangles (Standards G.SRT.6�8).
• Access Ramp - Student Task
This task has students design an access ramp, which complies with the Americans with Disabilities Act (ADA) requirements and include pricing based on local costs.
• Applications of the Pythagorean Theorem
This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review.
• Applications of the Pythagorean Theorem video
This video introduces and explains the topic.
This task can be used as a classroom activity. There is a lot of opportunity to discuss the process of mathematical modeling. It serves to illustrate MP 4 - Model with Mathematics, not just by engaging in the practice, but also by investigating what this practice entails.
• Bank Shot
This task asks students to use similarity to solve a problem in a context that will be familiar to many, though most students are accustomed to using intuition rather than geometric reasoning to set up the shot.
• Calculating Volumes of Compound Objects
This lesson unit is intended to help educators assess how well students solve problems involving measurement.
• Coins in a circular pattern
This task is intended for instructional purposes as an interesting activity which could accompany the other ''Seven Circles'' tasks. If it precedes these tasks, then the focus should be on recording information and looking for patterns.
• Congruence of parallelograms
Triangle congruence criteria have been part of the geometry curriculum for centuries. For quadrilaterals, on the other hand, these nice tests seem to be lacking. This task addresses this issue for a specific class of quadrilaterals, namely parallelograms.
• Congruent and Similar Triangles
The goal of this task is to understand similarity as a natural extension of congruence.
• Constructing Special Angles
The goal of this task is to estimate the measure of angles in triangles with integer side lengths.
• Defining Trigonometric Ratios
The purpose of this task is to use the notion of similarity to define the sine and cosine of an acute angle.
• Dilating a Line
This task asks students to "Verify experimentally" that a dilation takes a line that does not pass through the center to a line parallel to the original line, and that a dilation of a line segment (whether it passes through the center or not) is longer or shorter by the scale factor.
• Evaluating Conditions for Congruency
This lesson unit is intended to help educators assess how well students are able to work with concepts of congruency and similarity, including identifying corresponding sides and corresponding angles within and between triangles. They will also identify and understand the significance of a counter-example, and prove and evaluate proofs in a geometric context.
• Extensions, Bisections and Dissections in a Rectangle
This task involves a reasonably direct application of similar triangles, coupled with a moderately challenging procedure of constructing a diagram from a verbal description.
• Finding the Area of an Equilateral Triangle
This task examines how to calculate the area of an equilateral triangle using high school geometry.
• Finding triangle coordinates
The purpose of this task is to use similar triangles in order to study the coordinates of points which divide a line segment in a given ratio.
• GEOMETRY - Similarity, Right Triangles, and Trigonometry (G.SRT) - Sec Math II Core Guide
The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics II - Similarity, Right Triangles, and Trigonometry (G.SRT).
• How far is the horizon?
The purpose of this modeling task is to have students use mathematics to answer a question in a real-world context using mathematical tools that should be very familiar to them. The task gets at particular aspects of the modeling process, namely, it requires them to make reasonable assumptions and find information that is not provided in the task statement.
• Inscribing and Circumscribing Right Triangles
This lesson unit is intended to help educators assess how well students are able to use geometric properties to solve problems.
• Introduction to Fractals: Infinity, Self-Similarity and Recursion
This lesson is designed to help students understand aspects of fractals, specifically self-similarity and recursion.
• Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems.
• IXL Game: Geometry: Similar Figures
This game will reinforce geometry dealing with similar figures by looking at side lengths and angle measures. This is just one of many online games that supports the Utah Math core. Note: The IXL site requires subscription for unlimited use.
• Joining two midpoints of sides of a triangle
This task is closely related to very important material about similarity and ratios in geometry.
• Miniature Golf - student task
This task requires students to redesign a miniature golf course to make it more challenging.
• Module 6: Similarity & Right Triangle Trigonometry - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 6, Similarity and Right Triangle Trigonometry, introduces the last of the transformations, dilation. A big idea of Module 6 is that two figures are similar if a sequence of rigid transformation and dilations exists that maps one figure onto the other.
• Module 6: Similarity & Right Triangle Trigonometry - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 6, Similarity and Right Triangle Trigonometry, introduces the last of the transformations, dilation. A big idea of Module 6 is that two figures are similar if a sequence of rigid transformation and dilations exists that maps one figure onto the other.
• Mt. Whitney to Death Valley
The purpose of this task is to engage students in an open-ended modeling task that uses similarity of right triangles, and also requires the use of technology (e.g., printed or electronic maps), thereby illustrating SMP 5 - Use Appropriate Tools Strategically.
• Neglecting the Curvature of the Earth
This task applies geometric concepts, namely properties of tangents to circles and of right triangles, in a modeling situation.
• Proofs of the Pythagorean Theorem
This lesson unit is intended to help educators assess how well students are able to produce and evaluate geometrical proofs.
• Pythagorean Explorer
This applet challenges the student to find the length of the third side of a triangle when given the two sides and the right angle.
• Pythagorean Theorem
In this lesson students will be able to use the Pythagorean Theorem to find side lengths of right triangles, the areas of right triangles, and the perimeter and areas of triangles.
• Setting Up Sprinklers
This task involves several different types of geometric knowledge and problem-solving: finding areas of sectors of circles, using trigonometric ratios to solve right triangles, and decomposing a complicated figure involving multiple circular arcs into parts whose areas can be found.
• Seven Circles I
This task is intended to help model a concrete situation with geometry.
• Seven Circles III
This task is intended for instructional purposes only. It provides an opportunity to model a concrete situation with mathematics.
• Shortest line segment from a point P to a line L
This is a foundational geometry task designed to provide a route for students to develop some fundamental geometric properties that may seem rather obvious at first glance. In this case, the fundamental property in question is that the shortest path from a point to a line meets the line at a right angle, which is crucial for many further developments in the subject.
The goal of this task is to study if the analogue of the AA criterion for similarity of triangles holds for different types of quadrilaterals.
• Similar Triangles
This task works toward establishing the AA criterion for similarity of triangles by providing a detailed sequence of transformations that moves one of the given triangles to the other.
• Sine and Cosine of Complementary Angles
The goal of this task is to provide a geometric explanation for the relationship between the sine and cosine of acute angles.
• Slope Criterion for Perpendicular Lines
The goal of this task is to use similar triangles to establish the slope criterion for perpendicular lines.
• Solving Geometry Problems: Floodlights
This lesson unit is intended to help educators assess how well students are able to identify and use geometrical knowledge to solve a problem.
• Solving Problems with Circles and Triangles
This lesson unit is intended to help educators assess how well students are able to use geometric properties to solve problems
• Solving Quadratic Equations: Cutting Corners
This lesson unit is intended to help educators assess how well students are able to solve quadratics in one variable.
• Squaring the Triangle
Students can manipulate the sides of a triangle in this applet in order to better understand the Pythagorean Theorem.
The Hopewell people were Native Americans whose culture flourished in the central Ohio Valley about 2000 years ago. They constructed earthworks using right triangles. In this task, the student will look at some of the geometrical properties of a Hopewell earthwork.
• Tangent Line to Two Circles
The purpose of this task is to use similar triangles, setting up a proportion in order to calculate a side length.
• Tangent of Acute Angles
The purpose of this task is to focus on studying values of tanx for special angles and conjecturing from these values how the function tanx varies when 0x<90.
• Tessellations: Geometry and Symmetry
Students can explore polygons, symmetry, and the geometric properties of tessellations in this lesson.
• Trigonometric Function Values
The goal of this task is to explore the relationship between sine and cosine of complementary angles for special benchmark angles.
• Unit Squares and Triangles
This problem provides an opportunity for a rich application of coordinate geometry. http://www.uen.org - in partnership with Utah State Board of Education (USBE) and Utah System of Higher Education (USHE).  Send questions or comments to USBE Specialist - Joleigh Honey and see the Mathematics - Secondary website. For general questions about Utah's Core Standards contact the Director - Jennifer Throndsen .

These materials have been produced by and for the teachers of the State of Utah. Copies of these materials may be freely reproduced for teacher and classroom use. When distributing these materials, credit should be given to Utah State Board of Education. These materials may not be published, in whole or part, or in any other format, without the written permission of the Utah State Board of Education, 250 East 500 South, PO Box 144200, Salt Lake City, Utah 84114-4200.