# Gr. HS Unit: Similarity, Proof and Trigonometry

**LESSON UNIT**

**LESSON PLANS**

**LESSON SEEDS**

- How Far Can You Go in a New York Minute? (DOCX)
- Exploring Dilations (DOCX)
- AA Similarity (DOCX)
- Conjecturing a Proportionality Theorem (DOCX)
- Using Similar Triangles to Discover Trigonometric Ratios (DOCX)
- Right Triangle Trig Practice Problems
- Handicap Ramp (DOCX)
- Indirect Measurement (DOCX)

### UNIT OVERVIEW

Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. They identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles to understand right triangle trigonometry, with particular attention to special right triangles and the Pythagorean Theorem.

**Essential Questions:**

- How is visualization essential to the study of geometry?
- How does the concept of dilation connect to the concept of similarity?
- How does geometry explain or describe the structure of our world?
- How does the concept of similarity help to solve problems?
- How can reasoning be used to establish or refute conjectures?
- What are the characteristics of a valid argument?
- What is the role of deductive or inductive reasoning in validating a conjecture?
- What facts need to be verified in order to establish that two figures are similar?

A question is essential when it stimulates multi-layered inquiry, provokes deep thought and lively discussion, requires students to consider alternatives and justify their reasoning, encourges re-thinking of big ideas, makes meaningful connections with prior learning, and provides students with opportunities to apply problem-solving skills to authentic situations.

##### Unit Lesson

Additional information such as Teachers Notes, Enduring Understandings,Content Emphasis by Cluster, Focus Standards, Possible Student Outcomes, Essential Skills and Knowledge Statements and Clarifications, and Interdisciplinary Connections can be found in this Lesson Unit.

##### Available Model Lesson Plans

The lesson plan(s) have been written with specific standards in mind. Each model lesson plan is only a MODEL - one way the lesson could be developed. We have NOT included any references to the timing associated with delivering this model. Each teacher will need to make decisions related ot the timing of the lesson plan based on the learning needs of students in the class. The model lesson plans are designed to generate evidence of student understanding.

This chart indicates one or more lesson plans which have been developed for this unit. Lesson plans are being written and posted on the Curriculum Management System as they are completed. Please check back periodically for additional postings.

CCSC Alignment: G.SRT.3, G.MG.1

Students will use the AA Similarity Theorem to find the height of various real world objects.

##### Available Model Lesson Seeds

The lesson seed(s) have been written with specific standards in mind. These suggested activity/activities are not intended to be prescriptive, exhaustive, or sequential; they simply demonstrate how specific content can be used to help students learn the skills described in the standards. Seeds are designed to give teachers ideas for developing their own activities in order to generate evidence of student understanding.

This chart indicates one or more lesson seeds which have been developed for this unit. Lesson seeds are being written and posted on the Curriculum Management System as they are completed. Please check back periodically for additional postings.

CCSC Alignnment: G.SRT.2, G.MG.3

Application: The activity in this lesson seed could be used as a partner activity. This should be completed after lessons on similarity.

In this lesson seed, students use proportions and similar figures to adjust the size of the New York City Subway Map so that it is drawn to scale. Students are asked to evaluate whether these changes are necessary improvements.

CCSC Alignnment: G.SRT.1, G.SRT.1a, G.SRT.1b

Investigation: The activities in this lesson seed could be used after introductory instruction on dilations. By completing the activities provided in this lesson seed, students should gain a better understanding of some of the relationships between the properties of the image and pre-image of adilation.

CCSC Alignnment: G.SRT.3

Investigation: The activity in this lesson seed could be used during the direct instruction portion of a lesson to develop the criterion for AA Similarity.

This activity introduces the concept that AA is a sufficient condition to guarantee similar triangles.

(Note: This method does not address Standard G.SRT.3 but it could be used as another means to establish the AA Similarity criterion.)

CCSC Alignnment: G.SRT.4

Investigation: The activity in this lesson seed could be used during the direct instruction portion of a lesson targeting the first theorem mentioned in G.SRT.4.

This lesson seed provides an opportunity for students to discover that a line parallel to one side of a triangle divides the other two sides proportionally.

CCSC Alignnment: G.SRT.6

Investigation: The activity in this lesson seed could be used during the direct instruction portion of a lesson targeting G.SRT.6. The results of this discovery activity leads to the definition of the trigonometric ratios.

Right Triangle Trig Practice Problems

CCSC Alignnment: G.SRT.8

Practice: This lesson seed provides an interactive way for students to reinforce their understanding of using Trigonometric ratios in solving right triangles.

CCSC Alignnment: G.MG.3

Application:The activity provided in this lesson seed would be used to provide a learning experience for students that would help them to build proficiency with Standard for Mathematical Practice #1 (Make sense of problems and persevere in solving them). This activity makes use of ratio and proportion and the Pythagorean Theorem.

The activity in the lesson seed sets up a situation where students need to determine the length of a handicap ramp so that cost is minimized.

CCSC Alignnment: G.SRT.8

Application: The activity provided in this lesson seed would be used to provide a learning experience for students that would help them to build proficiency with Standard for Mathematical Practice #4 (Model with Mathematics).