Using the Pythagorean Theorem to Solve by Diagram

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Learn to apply the conceptual understanding of when and why the Pythagorean Theorem is equivalent in order to solve visually for missing side lengths

Big Idea

Moving students from hands-on Pythagorean Theorem to drawing diagrams to solve for missing sides continues the concept of why and how the equality works.

Bell Ringer

10 minutes

Now is a good time to discuss the concept of area of squares and length of sides on the square (the dimensions of the square).  Using a bellringer to remind students of the relationship between area and dimensions will be useful for beginning the solving by diagram activity today.  Therefore use the following questions as a bellringer to make the connection between area of squares and dimensions.


Again, solving by diagram activity today will really build from their understanding of how to calculate area squares and find dimensions of a square.




Wrapping up the Major Ideas

20 minutes

Open the PowerPoint of the book What’s Your Angle Pythagoras.  Turn the PowerPoint to slide number 14 and remind the students that yesterday was about figuring out when this relationship holds true.  Put the Poster up in the front of the room and ask students to take one minute to look at the poster and think other own about any patterns they notice with the triangles taped to the three categories of a2 + b2 as either equal, greater than, or less than the area of c2.  Then allow students one minute to discuss ideas with partner and be ready to share out important observations with the class.  This strategy is an example of the literacy strategy called Think-Pair-Share.  Script student observations on the board during the sharing time; if you are unsure of what it means to script, click on the link below to watch a short video about this strategy.

Scripting Strategy

Observations you are looking to surface:

  1. Equivalence only happens when the triangles are right triangles
  2. Inequality occurs with other types of triangles such as acute is greater than and obtuse were always less than

Now is a good time to bring up your pictures from the previous day.  If you up loaded these to a website and sorted each into before and after by type of triangle then it is easy to quickly show all the right triangles before and after to prove the equality of area.  You can then show the before and after of the acute triangles and see all the left-over tiles that mean greater than, and the obtuse images have an obvious section missing in the after photos. 

Additional Concept Building Material

Show the Pythagorean theorem water demo video of the water demonstration of the Pythagorean Theorem.  The diagram activity is about to ask students to think about moving the area from the two small squares into the largest square but also to move the area from the large square into separate small squares.  This concept of also moving area backwards to separate into two smaller squares is the focus of the water video.   Pause the video after the girl turns the wheel and all the water drains into the large square.  Discuss with students what is being represented (area of squares) and what is really being used in the tanks (volume).  Ask, “If the girl turns the wheel back over, what will happen?”  Show the video after your discussion.  After the video, ask, “Who can explain the different ways that the water can be shifted and drained to represent the shifting of the area between squares?”  If volunteers are scares, allow students one minute to discuss this question within cooperative groups and then ask again.  Script any important information on the board.  

Beginning the New Activity

25 minutes

Clarify your learning intensions with the students before beginning the next activity.  Inform students that the goal of understanding this equivalence when using right triangles can be used to find the length of missing sides on a right triangle – like in the book when Pythagoras used the formula to calculate the length of the ladder for Saltos and Pepros.  Today, we will begin to look at “how” to apply the equivalence to solve for missing side lengths on a right triangle.  You may even want to script this sentence on the board for students to reference.  

 Pass out the activity Solving the Pythagorean Theorem by Diagram and allow students to work in cooperative groups to answer questions 1 – 5.  As students work on these questions together, move about the room formatively assessing your students and providing feedback that moves their learning forward.  Through cooperative groups and your diligence, students will be provided many avenues of support to move their learning forward as they take ownership of their own learning. 

 After allowing students ample time (about 10 minutes) to complete the given tasks pull the class together for a mini wrap-up of the key ideas just discovered by answering each question.  Script any important information on the board and it is better to allow students to lead the wrap-up presentation. 

After the wrap-up, if time allows, ask students to begin solving for missing side lengths using the diagrams in question 6.  I did not assign these for homework yet because many students were still just on the edge of understanding how all the different ideas fit together and to work alone would have been counter-productive.  Unfinished examples were worked through in cooperative groups during the following class period. 

Standards in this Lesson

The math standard addressed by this lesson is 8.G.B.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

By using activities that allow students to take ownership of their own learning through cooperative feedback the first math practice standard is used:

MP1 Make sense of problems and persevere in solving them.

Applying the strategy of mini-wrap ups that are student centered will directly bring math practice standard 3 into the lesson. 

Math practice standard seven is incorporated as students are asked to use area of squares, shifting the area of squares, and then finding the dimensions of a square based on area in order to solve Pythagorean Theorem Problems. 

MP7 Look For and Make Use of Structure.