Proving Pythagorean Theorem Using Similar Triangles

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Objective

SWBAT use similar triangles to prove the Pythagorean Theorem.

Big Idea

Altitude Adjustment? By strategically introducing the altitude to the hypotenuse of a right triangle, students will be able to prove the Pythagorean Theorem.

Activating Prior Knowledge

5 minutes

In today's lesson, my students will be proving the Pythagorean Theorem using similarity. Students should have had experience applying the Pythagorean Theorem in 8th grade, but I want to make sure that they know the Theorem before we prove it.

Teacher's Note: This lesson can be taught as a Block period or broken up into two lessons.

So as an entry into the lesson I have my students work on the Pythagorean Theorem Warmup, which is a problem involving the slope of an access ramp. I will have my students seated in pairs for the entirety of this lesson. I want students who need help will be able to get it without relying solely on me.

I give my students 2-3 minutes to work on the Warmup problem. As Warmup time is ending, I will walk around quickly to see how many of my students have arrived at the correct answer of 7/24. I haven't allotted a lot of time to stop and consult, so if a student is stuck, I quickly try to partner them with a nearby student who can help them.

When I have given the room a once over, I will state the learning goals explicitly to the class. I might say something like:

Class, as you know, we've been studying similarity and proving geometric relationships using triangle similarity. The Pythagorean Theorem is one of the most famous (and most proven) theorems in mathematics.

Did you know that there are at least 367 different ways to prove the Pythagorean Theorem?

Today we are going use similar triangles to prove the Pythagorean Theorem [I write this on the board]. We will take a seemingly long and circuitous journey to get to the proof, but keep in mind all along that our goal is to use similar triangles to prove the Pythagorean Theorem.

Hands-On Experience

20 minutes

In this section of the lesson my students will be working on Constructing the Altitude to the Hypotenuse of a right triangle. My intention is that they will convince themselves that this creates two right triangles that are similar to the original right triangle and to each other. The materials students will need are:

  • Compass
  • Straightedge 
  • Scissors
  • Plain White Paper 

To make sure that my students are prepared to construct the altitude to the hypotenuse, I will:

  1. Demonstrate the construction on page 1: the perpendicular to a line through a point not on the line.
  2. Define altitude and provide diagrams to illustrate how the altitude can be inside, outside, or on the triangle.

Next, I will have my students turn to Page 2 and construct the altitude to the hypotenuse. As they are performing this construction, I will walk around handing out one sheet of plain white paper to each student (for tracing). As I'm passing the papers out, I will also be checking to make sure that my students are performing the constructions correctly. Common mistakes on this construction include constructing the perpendicular bisector or median to the hypotenuse. Also, some students need clarification on what the direction "to the hypotenuse" means in this instruction.

As my students complete the construction and move on to the copying and cutting part of the activity, they often want explicit directions regarding what they should cut. I point out that the prompt asks them to determine relationships between triangles. I also share that the purpose of cutting is to enable them to physically manipulate the triangles so that they can compare them. I then ask, "What triangles do you need to compare?"

As I walk around the room when I see a students manipulating the triangles, I may stop and say, "So, show me what you're discovering about the relationships between the triangles". I want to motivate students to use the triangles as a tool for exploration. When I encounter a student who seems to be stagnating, I will encourage him/her to pick two triangles to compare. Then, directing them to take those triangles into their hands, I'll say "Show me how you can manipulate these triangles to determine some relationships between them."

With this type of direction and with additional help from peers, I expect that most of my students will arrive at the conclusion that all three triangles are similar by the AA Similarity Postulate.

Some students will want to write formal statements and justifications at this point. This is ok, but it's not required. They will have the chance to formalize their findings in the next section. The objective of this Launch activity is met when students convince themselves that all three triangles are similar.

 

 

 

Similarities and Correspondences

15 minutes

Whereas the previous section was geared toward student discovery, and informal understanding, in this section I want my students to formally explain why the three triangles are similar. For that reason, I scaffold this activity to ensure more direct progress towards a generalization.

I start by giving my students Seeing the Similarities. I ask them to complete and justify the first two similarity statements using the sentence frames:

  1. ^ DBC ~ ^ ________ by ____________ because... 
  2. ^ DBC ~ ^ ________ by ____________ because... 

I will demonstrate the possible justifications for the last pair of triangles because it requires reasoning that I would not expect most of my students to arrive at on their own.

The following video presents the demonstration that I like to give at this stage of the lesson:

 

An alternate justification for the two smaller triangles being similar, which I also provide, is that similarity is transitive. Since both triangles are similar to the larger triangle DBC, we can conclude that they are similar to each other by transitive property of similarity. I like this justification because it reinforces the concept of transitivity, an important concept in Geometry.

In any case, once we have finished this portion of the lesson together, I will have my students work independently on the next page, which should be fairly self-explanatory.

Sorting it all Out

30 minutes

At this point in the lesson, I'm gradually releasing control to the students as they make their way toward the final stretch. But FIRST, I really need to make sure that they are clear on how they will use the Pythagorean Theorem Card Sort Protocol. I take some time to ready through the document with the class, stopping to clarify and emphasize as needed. My main points of emphasis are:

  1. Keep your workspace clutter-free.
  2. For the benefit of you and your partner, follow the protocol as it is written.
  3. Own your creative control and be strategic in exercising it.
  4. Keep your mind on the goal. 

Once I have briefed students on the protocol and the mission, give each pair of students the Pythagorean Proof Equation Bank and the True Equations Table, and they are free to work.

I walk around making sure that each group is adhering to the protocol, and moving students forward that appear to be stuck or having misconceptions.

As pairs of students claim that they are ready to paste their cards on, I ask them to show me how they've exercised their creative control in order to communicate what they know and how they know it. One of the basic things I check for is whether the students have make explicit connections between the means-extremes equations and the proportions that implied them. Also there is a true means-extremes equation h^2=xy that doesn't have a correspondents proportion equation. I check to see whether students have used their black to make the proportion. There is an underlying message here that says if something that should exist doesn't exist, then create it.

 

 

Writing the Proof

30 minutes

Now, my students have done all of the preparation work necessary to prove the Pythagorean Theorem using similarity. In this section I will give them some additional mathematical tools to write the proof. I frame this proof writing task with a discussion of identities:

  1. I define an identity as an equation stating that two expressions are equivalent for all possible values of their variables.
  2. I explain that in math we often want to prove that an identity is actually an identity, and that we have techniques for doing so. The basic technique, I tell them, is to perform a series of "mindful manipulations" on one side of the identity until we can get the left side and right side to be "identical". As an illustrative example I demonstrate the proof that (a+b)(a-b)=a^2-b^2 is an identity.
  3. I clarify that we are attempting to prove that the Pythagorean Theorem is an identity as long as a and b are leg lengths of a right triangle and c is the hypotenuse. In other words, given our arbitrary right triangle (from the diagram), we must show that a^2 + b^2 = c^2 is an identity.
  4. I remind students that in the previous activity they have sorted out the true equations related to the diagram. I say, "These will be useful in performing the mindful manipulations they will need to perform.

At this point, I allow students to have time to struggle and solve the problem. I love to see the light bulbs turning on as students' effort and perseverance start to pay off. As students begin to write their proof I remind them that our hypothesis is for an arbitrary right triangle with legs a and b, and hypotenuse c. I say, "We must use this as the starting point and from this statement prove the conclusion a^2 + b^2 = c^2 follows."

As students work I will conference with them about their approach. When I do so, I expect that they can explain the flow of the proof by saying something like:

  1. Show that the altitude to the hypotenuse creates similar triangles
  2. Similar triangles can be used to write proportions that lead to equations for a^2 and b^2
  3. Algebraic manipulations can be performed to show that the Pythagorean Theorem is an identity.

Presentations

30 minutes

To bring the lesson to closure, I will select students to present. I have two different criteria in mind for selecting presenters:

  1. The first type of student paper I select demonstrates what I would like to see in terms of establishing a sound line of reasoning beginning with an arbitrary right triangle and concluding with a^2+b^2=c^2. These students' presentation will focus on the layout and sequence of steps in their proof. In other words, these presentations will focus on the big picture.
  2. The second type of paper I select is chosen based on the particular approach it employed to show that a^2+b^2=c^2 is an identity. Invariably, some students select a very simple and elegant approach while others select more algebraically impressive paths. These students' presentations will focus on the algebraic reasoning used in their particular approach.

My instructional goal is to get students to see that various solution paths are possible and they all have their own merits. See Student Work_prove PT using similarity to see the types of exemplars that I have selected in the past.