Primary Type: Lesson Plan
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 71771
Altitude to the Hypotenuse
In this lesson, students discover what happens when the altitude to the hypotenuse of a right triangle is drawn. They learn that the two triangles
created are similar to each other and to the original triangle. They will learn the definition of geometric mean and write, as well as solve, proportions
that contain geometric means. All discovery, guided practice, and independent practice problems are based on the powerful altitude to the
hypotenuse of a right triangle.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Suggested Technology: Document Camera, LCD
Projector, Adobe Acrobat Reader, Microsoft Office
Instructional Time: 50 Minute(s)
Resource supports reading in content area: Yes
Freely Available: Yes
Keywords: altitude, complementary angles, geometric mean, hypotenuse, proportions, right triangle, similarity
Resource Collection: FCR-STEMLearn Geometry
ATTACHMENTS
Altitude to the Hypotenuse Warm Up.docx
Altitude to the Hypotenuse Warm Up.pdf
Altitude to the Hypotenuse Notetaking Worksheet.docx
Altitude to the Hypotenuse Notetaking Worksheet.pdf
Altitude to the Hypotenuse Notetaking Worksheet ANSWERS.pdf
Altitude to the Hypotenuse Practice Worksheet.docx
Altitude to the Hypotenuse Practice Worksheet.pdf
Altitude to the Hypotenuse Practice Worksheet ANSWERS.pdf
LESSON CONTENT
Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
The student will understand and be able to:
1. Construct the altitude to the hypotenuse in a right triangle.
2. Discover similar relationships in right triangles.
3. Apply similar relationships in right triangles to solve problems.
Prior Knowledge: What prior knowledge should students have for this lesson?
Students should:
1. Know the definition of complementary angles.
2. Know how to find the measure of one acute angle of a right triangle given the other.
3. Know the definition of similarity, especially as it relates to triangles.
4. Know the ways to prove triangles are similar, AA~ Postulate, SAS~ Theorem, and SSS~ Theorems.
5. Know how to set up similarity proportions for lengths of corresponding sides of similar triangles.
page 1 of 3 6. Know how to write and solve proportions, especially given a pair of similar triangles.
7. Be familiar with or introduced to a variety of tools (compass, straightedge, string, reflective devices (mirrors or MIRA), folded paper (patty paper), dynamic
geometric software, etc.) used to make formal geometric constructions.
8. Be familiar with or introduced to basic vocabulary and techniques used to make formal geometric constructions.
To activate prior knowledge, teacher can ask students to solve a few simple proportions, where in at least one of the proportions the second and third term are equal,
(e.g. at least one proportion has a geometric mean).
Guiding Questions: What are the guiding questions for this lesson?
1. What is the relationship between the angles formed when the altitude to the hypotenuse of a right triangle is formed?
2. What can you conclude about the triangles formed when the altitude to the hypotenuse is constructed/drawn?
3. Can you explain why or why not this works for a triangle that is not a right triangle?
4. Given a diagram where the altitude to the hypotenuse is drawn, name the three triangles that are similar to one another.
5. What segments are proportional? Why?
6. Using a previous or similar diagram, can you write and solve an extended proportion using all pairs of corresponding segments?
7. Justify why you are able to write the extended proportion.
Teaching Phase: How will the teacher present the concept or skill to students?
The teacher will distribute Altitude to the Hypotenuse Notetaking Worksheet, and direct students to complete Discovery Problem Number 1 in pairs.
The teacher should circulate around the room and provide clarification and assistance where needed.
Students should be directed to work through the discovery problem, and then during debriefing and consensus, students should be directed to write the words for
the theorem in the dotted box on the paper. (See Altitude to the Hypotenuse Notetaking Worksheet Answer Key).
Follow-up with whole class discussion, debriefing, and consensus. (Teacher can use GeoGebra or other dynamic geometry software for illustrations purposes for
any or all problems within the worksheet or lesson).
The teacher will take note of skill gaps, misconceptions, and specific difficulties so they may be addressed throughout the lesson. (Additional example(s) may be
given as necessary).
At the conclusion of Discovery Problem Number 1, students should be able to:
State, either formally or informally, the theorem: When the altitude to the hypotenuse of a right triangle is drawn, the two triangles created are similar to each
other and to the original triangle.
Write a similarity statement relating the three triangles when the altitude to the hypotenuse of a right triangle is drawn.
The teacher will direct students to complete Discovery Problem Number 2 in pairs.
The teacher should circulate around the room and provide clarification and assistance where needed.
Students should be directed to work through the discovery problem, and then during debriefing and consensus, students should be directed to write definition of
geometric mean and the words for the theorems in the dotted boxes on the paper. (See Altitude to the Hypotenuse Notetaking Worksheet Answer Key).
Follow-up with whole class discussion, debriefing, and consensus. (Teacher can use GeoGebra or other dynamic geometry software for illustrations purposes for
any or all problems within the worksheet or lesson).
The teacher will take note of skill gaps, misconceptions, and specific difficulties so they may be addressed throughout the lesson. (Additional example(s) may be
given as necessary).
At the conclusion of Discovery Problem Number 2, students should be able to:
Define and give an example of geometric mean.
State, either formally or informally, the corollary: When the altitude to the hypotenuse of a right triangle is drawn, the length of the altitude is the geometric mean
between the two segments of the hypotenuse.
Write and solve a proportion where the altitude to the hypotenuse is the geometric mean between the two segments of the hypotenuse.
State, either formally or informally, the corollary: When the altitude to the hypotenuse of a right triangle is drawn, each leg is the geometric mean between the
entire hypotenuse and the segment of the hypotenuse adjacent to it.
Write and solve proportions where each leg is the geometric mean between the entire hypotenuse and the segment of the hypotenuse adjacent to it.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
The teacher will distribute Altitude to the Hypotenuse Practice Worksheet, and direct students to complete problem 1 - 5 in pairs.
The teacher should circulate around the room and provide clarification and assistance where needed.
Follow-up with whole class discussion, debriefing, and consensus. (Teacher can use GeoGebra or other dynamic geometry software for illustrations purposes for
any or all problems within the worksheet or lesson).
The teacher will take note of skill gaps, misconceptions, and specific difficulties so they may be addressed throughout the lesson.
Altitude to the Hypotenuse Practice Worksheet ANSWERS
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the
lesson?
As students finish numbers 1 - 5 in pairs, they can be directed to complete number 6 - 10 independently.
Students will submit a written assignment of the teacher's choice, and Altitude to the Hypotenuse Practice Worksheet.
(Optional Altitude to the Hypotenuse Homework Worksheet is attached here).
(Optional Altitude to the Hypotenuse Homework Worksheet ANSWERS is attached here).
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
The teacher will pose the question, "What three things happen when the altitude to the hypotenuse is drawn for any right triangle?" (as investigated and described in
Altitude to the Hypotenuse Notetaking Worksheet) . Students may respond in either oral or written form.
At closure, all students should have their Altitude to the Hypotenuse Practice Worksheet completed.
Summative Assessment
The teacher will administer assessment titled Altitude to the Hypotenuse Daily Quiz, where students demonstrate mastery of writing a similarity statement containing
three similar triangles, given a triangle where the altitude to the hypotenuse is drawn; writing and solving a proportions where the altitude is the geometric mean
between the two segments of the hypotenuse, and writing and solving a proportions where each leg is the geometric mean between the entire hypotenuse and the
page 2 of 3 segment of the hypotenuse adjacent to it.
Please Note: The Altitude to the Hypotenuse Daily Quiz contains six homework questions and is designed to be used as a warm-up for the next day.
Altitude to the Hypotenuse Daily Quiz ANSWERS
Formative Assessment
This lesson is designed as a part of a unit on similarity of triangles, and should follow lesson(s) on ways to prove triangles are similar. For this lesson , the teacher
should provide a warm-up, to assess and revive students' prior knowledge of finding the measure of one complementary angle given the other, writing similarity
statements for two similar triangles, as well as writing extended proportion for all pairs of corresponding sides.
Altitude to the Hypotenuse Warm Up
Altitude to the Hypotenuse Warm Up ANSWERS
Teacher will do comprehension checks as direct instruction is given on Altitude to the Hypotenuse Notetaking Worksheet.
Student Altitude to the Hypotenuse Activity Practice Worksheet will allow more formative assessment by the teacher. Students may peer-check as well during this
activity.
Altitude to the Hypotenuse Notetaking ANSWERS
Feedback to Students
Students will receive feedback via internal, oral, and written methods throughout the lesson from themselves, the teacher, and classmates as follows:
Altitude to the Hypotenuse Notetaking Worksheet: self, (internal), classmate, (oral), or, teacher, (oral), during, and at completion of activity.
Altitude to the Hypotenuse Practice Worksheet: self, (internal), and teacher, (oral), during, and at completion of activity.
(Optional) Altitude to the Hypotenuse Homework Worksheet: self, (internal), or teacher, (oral), during, or (written), at completion of activity.
Altitude to the Hypotenuse Daily Quiz: teacher, (written), at completion of assessment.
(See Attachments for all documents referenced in this portion of the lesson plan).
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations:
For students with special needs, or lower performing students, use strategic pairing during the discovery problems, and adjust the amount of practice problems and/or
homework problems.
Suggested Technology: Document Camera, LCD Projector, Adobe Acrobat Reader, Microsoft Office
Additional Information/Instructions
By Author/Submitter
This lesson aligns with the following standard for mathematical practice:
MAFS.K12.MP.7.1 - Look for and make sense of structure
SOURCE AND ACCESS INFORMATION
Contributed by: Cynthia Higgins
Name of Author/Source: Cynthia Higgins
District/Organization of Contributor(s): Palm Beach
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.G-SRT.2.5:
Description
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Remarks/Examples:
Geometry - Fluency Recommendations
Fluency with the triangle congruence and similarity criteria will help students throughout their investigations of
triangles, quadrilaterals, circles, parallelism, and trigonometric ratios. These criteria are necessary tools in many
geometric modeling tasks.
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