Lesson Title: Prove Pythagorean Identities
Date: _____________ Teacher(s): ____________________
Course: Common Core Algebra II, Unit 7
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?
F.TF.C.8 Prove the Pythagorean identity sin2(t) + cos2(t) = 1 and use it to find sin(t), cos(t), or tan(t), given sin(t), cos(t), or
tan(t), and the quadrant of the angle.
MP2: Reason abstractly and quantitatively.
MP3: Construct viable arguments and critique the reasoning of others.
MP5: Use appropriate tools strategically.
Lesson Launch Notes: Exactly how will you use the
first five minutes of the lesson?
Find all the missing angles and sides of the given right
triangles.
Use the lesson launch document.
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.
Explain why the ratio
cosA 
7
3
is not a possible ratio
for a right triangle.
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.
2.
3.
4.
5.
6.
Have students work together to complete the lesson launch activity. They may use the Trig Table or their
calculator to determine the missing side values of each right triangle. Recommend extending the decimal a few
places for accuracy. Once complete, have students participate by either drawing the triangles on the board for
students to show their work, or using a document camera for students to show their work to other students.
(Look for evidence of MP5.)
Ask students “Which formulas can you recall for triangles?” Have a volunteer list them on the board. If no one
mentions the Pythagorean Theorem, ask them, “What about Right Triangles?” Hopefully someone will mention
a2 + b2 = c2. Ask, “What does the Pythagorean Theorem help us find?” Once the discussion has ended, relating
the theorem to missing side values will show the students the purpose for today. (Look for evidence of MP3.)
Refer students to the lesson launch problems. Ask them to work together to prove that each triangle is a right
triangle using only the sides they calculated. Ask students what they notice about each triangle and what was
used to find each side length. Ask students to get into pairs and create their own right triangle with a hypotenuse
of 1 unit. Let them determine the angle measures and side lengths. Once they have created their triangles have
the pairs get into groups of 4 to share out. Try to lead the discussions to help students recognize that sin(A) and
cos(A) represent the opposite and adjacent sides of the right triangle. (Look for evidence of MP2.)
Provide students with the activity sheet and ask them to use the Pythagorean Theorem, and the Trigonometric
Ratios to prove the Pythagorean Identity. Allow students to work in groups. Walk around and answer questions
the groups might have. Try to lead the groups towards using substitution. Refer to the activity key to see one
method to prove the Identity. Once students have completed their proof, allow students time to share with the
class as to how they proved the Pythagorean Identity. (Look for evidence of MP3.)
Provide students with the practice activity. First, have students work individually to complete their own
problem. Once each member has completed the problem, they should switch papers with a partner, check the
other person’s solution and help make corrections as needed. Once both partners agree, they can move on to the
next problem. Circulate around the room and allow students to ask questions. See how they are working through
the problems and challenge them to try different methods to find the missing value. (Look for evidence of MP2
and MP3.)
Once the students have completed the practice sheet, present the lesson closure and ask students to determine if
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: Prove Pythagorean Identities
Date: _____________ Teacher(s): ____________________
it is possible. Provide students with the homework.
Course: Common Core Algebra II, Unit 7
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 will know the Pythagorean Identity, where it comes from, why it works, and how to apply it to find other
trigonometric ratios.
Notes and Nuances: Vocabulary, connections, anticipated misconceptions (and how they will be addressed), etc.
This is designed to be a multi-day lesson. A good potential stopping point for day one could be after they are asked
to prove the identity. If your class is moving a little faster, you can continue and stop where you feel best suits your
class. To complete the activity, students need to have learned the tan(A) ratio. Tan(A)=sin(a)/cos(A). They may need
a review on how the quadrant affects the value of sin(A), cos(A), and tan(A).
If students need support during the lesson launch, ask them if they can recall the trigonometric ratios from geometry
and how they work. The purpose of keeping the hypotenuse of 1 is to lead towards the Pythagorean identity. The
lesson launch does not need to be given out as a worksheet. It can be placed under a document camera or woven into
a PowerPoint.
When asking for formulas for triangles, students will provide a few different examples. The purpose for the openended question is to allow the students to recall the Pythagorean equation without it being given to them.
Students need to understand that there is more then one way for the students to find the sin(A), cos(A), and tan(A).
Allow flexibility, but hopefully some will also use the Pythagorean Identity to find the missing value.
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?
Lesson Launch Document
Activity Document
Practice Document
Scientific/Graphing Calculators
Textbooks have ample problems that can be used. You
may need to modify the directions to match the lesson.
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?
Are students able to use the Pythagorean identity to find other ratios?
Can students recognize how different quadrants change values in those ratios?
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this
product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.