Transcript Similar Triangles - Pascack Valley Regional High School

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 (to
introduce 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 applying
theorems.
• 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
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