Lesson Title: Discovering the Formula for the Area of a Triangle Course: ___________________________
Date: _____________ Teacher(s): ____________________
Start/end times: _________________________
Lesson Standards/Objective(s): What mathematical skill(s) and understanding(s) will be developed? Which
Mathematical Practices do you expect students to engage in during the lesson?
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G.SRT.9(+) Derive the formula A  ab sin( c) for the area of a triangle by drawing an auxiliary line from a vertex
perpendicular to the opposite side.
MP1: Make sense of problems and persevere in solving them.
 and quantitatively.
MP2: Reason abstractly
MP3: Construct viable arguments and critique the reasoning of others.
Lesson Launch Notes: Exactly how will you use the
first five minutes of the lesson?
Provide students with the Find that Area warm up.
Using what they learned in the Can we Build It? Task,
have students find the area of the park. Display student
work up from the previous day’s task, so students have
something to refer to. Look for MP1 as students work
independently. Then, as a class discuss your solutions.
Lesson Closure Notes: Exactly what summary activity,
questions, and discussion will close the lesson and
connect big ideas? List the questions. Provide a
foreshadowing of tomorrow.
Have students partner up and ask them to answer the
following question together:
When would it be useful to have the new area formula we
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developed today A  ab sin( c) ? When would you use
the traditional are of the triangle formula?
Possible Answer: If you are given the two sides and an
angle
and you don’t have the height, this formula would
be helpful. You would use the traditional formula when
you are given a right triangle or when you are given the
base and the height measurements.
Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities, investigations,
problems, questions, or tasks will students be working on during the lesson? Be sure to indicate strategic
connections to appropriate mathematical practices.
1. Place students in groups of 3-4. Explain that today they will be creating a formula they can use to find the area
of a triangle when only given two adjacent sides and an included angle. Provide them with a copy of the
Deriving the Area of the Triangle diagram and have them work with their groups to develop a formula. Look for
evidence of MP #2 and 3.
2. As students develop their formula, provide them with dynamic geometry software, such as, Geometer’s
Sketchpad or GeoGebra. Provide them with a copy of the G.SRT.D.9 Sketchpad Resource Sheet for basic
sketchpad directions, so students are able to test their conjecture for all different types of triangles. (Refer to the
G.SRT.D.9 Sketchpad Area of a Triangle Formula video for an example.)
3. After students test their examples, have groups share out the different formulas that they developed and tested,
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even if they are not the traditional A  ab sin( C) . (Example: Students may develop a formula
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A  bc sin( A) , etc.) As a class, discuss how they are similar and different and eventually come to the
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traditional formula A  ab sin( C) , but allow students to use any version of the formula they choose.
2

HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.

Lesson Title: Discovering the Formula for the Area of a Triangle Course: ___________________________
Date: _____________ Teacher(s): ____________________
Start/end times: _________________________
Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how will I
measure student success? That is, deliberate consideration of what performances will convince you (and any outside
observer) that your students have developed a deepened and conceptual understanding.
Students should be able to develop a version of the area of the formula of the area of a triangle using sine and test
several different types of triangles to know if their conjecture holds true.
Notes and Nuances: Vocabulary, connections, anticipated misconceptions (and how they will be addressed), etc.
Connections should be made to pervious understandings of trigonometric ratios, specifically sine. Students should
also already be familiar with the basic area of a triangle.
Resources: What materials or resources are essential
for students to successfully complete the lesson tasks or
activities?
Homework: Exactly what follow-up homework tasks,
problems, and/or exercises will be assigned upon the
completion of the lesson?
Find that Area warm up
Deriving the Area of the Triangle
Mobile or computer labs with dynamic geometry
software, such as Geometer’s Sketchpad or GeoGebra
G.SRT.D.9 Sketchpad Resource Sheet
G.SRT.D.9 Sketchpad Area of a Triangle Formula
Video
Have students create two triangle diagrams where the new
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area formula ( A  ab sin( C) ) would be appropriate to
use and two triangle diagrams where the traditional area
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formal ( A  bh ) is more appropriate to use. Also, have
 provide an answer key for their diagrams, so
students
students can try each other’s problems the next day as a
warm up.

Lesson Reflections: How do you know that you were effective? What questions, connected to the lesson
standards/objectives and evidence of success, will you use to reflect on the effectiveness of this lesson?
Were students able to develop and test their conjectures for the area of a triangle using sine?
Are students able to use this formula to find the area of a triangle?
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Are students able to articulate the difference between solving problems with the standard area formula, A  bh ,
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and the new area formula, A  ab sin( C) ?

Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a
Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Creative Commons
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.