Unit 2, Lesson 16
Proving the Pythagorean Theorem
Integrated Math 2
16.1
Warm-up
Instructional Routines
Notice and WonderMaterials
NoneActivity Narrative
The purpose of this Warm-up is for students to get familiar and comfortable with the diagram of an altitude drawn to the hypotenuse of a right triangle with side lengths labeled with variables, which will be useful when students write equations using these variables and manipulate them in a later activity. While students may notice and wonder many things about this image, the similarity of the triangles and relationships among the side lengths are the important discussion points.
By engaging with this explicit prompt to take a step back and become familiar with a context and the mathematics that might be involved, students are making sense of problems (MP1).
Launch
Arrange students in groups of 2. Display the image. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things they notice and wonder.
Activity
NoneWhat do you notice? What do you wonder?
Student Response
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Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If similarity does not come up during the conversation, ask students to discuss this idea. The following activity uses the same diagram, so students do not need to resolve all of the questions they are wondering about at this point.
Lesson Synthesis
Add the following theorem to the class reference chart, and ask students to add it to their reference charts:
Pythagorean Theorem: If a right triangle has legs with lengths and
and hypotenuse with length , then . (Theorem)
Ask students to brainstorm the kinds of problems that the Pythagorean Theorem helps them solve.
Ask students why it might be useful to prove the same thing in multiple ways. (If one way is confusing or relies on something someone doesn’t understand, then it can be good to have another way to prove it.) Then ask students to consider times in their life when it has been useful to make a point in a conversation by approaching that point in several different ways.
Student Lesson Summary
In any right triangle with legs and and hypotenuse , we know that . We call this the Pythagorean Theorem. But why does it work?
We can use an altitude drawn to the hypotenuse of a right triangle to prove the Pythagorean Theorem.
Right triangle labeled A B C. Right angle points up. Left side labeled A, right side labeled b, arrow on bottom side labeled c. Altitude drawn from right angle to bottom side, labeled f, creating 2 segments on bottom side, left side labeled d creating triangle F D A, right side labeled e, creating triangle F E B.
We can use the Angle-Angle Triangle Similarity Theorem to show that all 3 triangles are similar. Because the triangles are similar, the corresponding side lengths are in the same proportion.
Because the largest triangle is similar to the smaller triangle, . Because the largest triangle is similar to the middle triangle, . We can rewrite these equations as and .
We can add the 2 equations to get that , or . From the original diagram we can see that , so , or .