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Similar Triangles

A High School Geometry Unit

by Mary Doherty

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Unit Objectives

• Students will be able to determine if two triangles are similar.

• Students will be able to find angle and side measures of similar triangles using congruence and proportions.

• Students will apply the properties of similar triangles to solve problems.

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NJ C.C.C.S.

• 4.2.Geometry and Measurement– A.1:Use geometric models to represent real-world situations and

objects and to solve problems using those models

– E.1: Use techniques of indirect measurement to represent and solve problems (similar triangles)

• 5.4 Mathematical Processes– F.5: Use computer software to make and verify conjectures about

geometric objects.

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Day 1

• Prerequisites: completed lessons on:– Ratios and proportions

– Similar figures

• Objectives– Students will use technology to explore the concept of similar

triangles.

– Students will make connections between similar figures and similar triangles and apply those connections to the properties of similar triangles.

– Students will be able to identify similar triangles using one of three postulates and theorems.

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Day 1

• Warm-up – students to read about similar figures at regentsprep.org to recall

attributes (and initiate connections)

• Instructional Strategy– students will work with a partner on SASinSchool InterActivity #890

(Triangles: Proving Similarity). Students to complete in-class worksheet. Teacher to serve as guide to facilitate achievement of objectives

• Homework– SASinSchool follow-up worksheet

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Day 2

• Objectives– Students will be able to identify similar triangles.

– Students will apply the properties of similar triangles to solve problems.

• Warm-up– Proving triangles congruent (to connect congruence short-cuts

(i.e., SSS, SAS, AAS, and ASA) to similarity short-cuts(i.e., AA, SSS, SAS).

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Day 2

• Instructional Strategies– Students to take notes on definitions of similar triangles and postulates

and theorems used to prove triangles similar. Examples included in notes. (overhead projector)

– Students to complete a brief problem set on determining whether two triangles are similar and, if so, stating the postulate or theorem used to prove this and writing a similarity statement.

– Students to do three problems on finding angle or side measures by setting up proportions. Students to write solutions on blackboard.

• Homework– worksheet from Teacher Resource workbook

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Day 3

• Objective – Students will use similar triangles and indirect measurement to

measure the heights of large objects.

• Warm-up– Students to view BrainPop

on similar triangles (tointroduce indirect measurement)

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Day 3

• Instructional Strategies– Brief note-taking on concept of indirect measurement using similar

triangles. Notes to include sketches and examples.

– Students will work outside in groups of three to collect data (measuring shadows) to calculate the height of various large objects, such as trees, basketball backboards, flagpoles, etc.

– After collecting data using the activity worksheet, students will return to the classroom and calculate the approximate height of these objects using similar triangles.

– Groups to compare results.

• Homework– worksheet of indirect measuring problems

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Day 4

• Objective– Students will use proportionality theorems to calculate lengths of sides

in triangles.

• Warm-up– Parallel line problems (parallel lines intersected by a transversal)

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Day 4

• Instructional Strategies– Geometer’s Sketchpad Lab Parallel Lines in a Triangle. Students to

construct sketches as directed and answer “discovery” questions.

– Student note-taking on proportionality theorems with examples.

– Guided practice solving problems applyingtheorems.

• Homework– Textbook assignment to practice applying

concepts

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Day 5

• Objective – Students will analyze how a ray bisecting an angle of a triangle divides

the sides of the triangle proportionally.

• Warm-up– Students to construct and label a triangle in Geometer’s Sketchpad with

a ray bisecting an angle to prepare for investigative activity.

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Day 5

• Instructional Strategies– Using the sketch constructed in the warm-up, students to measure sides

of each the two triangles formed by the bisector. Students will then calculate ratios of sides to determine which corresponding parts are proportional.

– Summary note-taking with examples

– Independent practice

• Homework– Worksheet from Teacher Resources

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Unit Project

• Objectives – Students to construct and explore

a “Golden Rectangle” and the Golden Ratio, a melding of art and history since ancient civilizations

– Students to research history of Golden Ratio (and Golden Rectangles and Triangles) and applications in art, architecture and nature

– Students to connect Sketchpad construction of the golden rectangle and golden spiral to its applications throughout the ages.

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Unit Project

• Project Summary– Technology: construction of golden rectangle and golden spiral.

Calculation of golden ratio.

– Research to identify applications of golden rectangle, triangle, ratio, or spiral in history, art, architecture, nature or other area of student interest.

– Communication: students to write a two-page essay to summarize findings and connect to construction.

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Assessment

• Informal– Daily homework

– Student questions

– Guided and independent practice

– Student conjectures during investigative activities

– Student blackboard work

• Formal– Geometer’s Sketchpad lab

reports and sketches

– SasInSchool interactivity investigation

– Indirect measuring activity summative results

– Golden Rectangle project

– Unit test