Similar Triangles

1. Similar Triangles A High School Geometry Unit by Mary Doherty

2. 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.

3. 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.

4. 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.

5. 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

6. 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).

7. 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

8. Day 3 • Objective • Students will use similar triangles and indirect measurement to measure the heights of large objects. • Warm-up • Students to view BrainPopon similar triangles (tointroduce indirect measurement)

9. 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

10. 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)

11. 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 applyingconcepts

12. 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.

13. 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

14. 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.

15. 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.

16. 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 Assessment