Pythagorean Theorem

Lesson by Joy Clubine, Alannah McGregor & Jisoo Seo
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Curriculum 

  • Students will determine the Pythagorean relationship, through investigation using a variety of tools and strategies.
  • Solve problems involving right triangles geometrically, using the Pythagorean relationship.

Summary 

These investigations are intended to deepen students’ understanding of the Pythagorean theorem. Previous exposure to the Theorem is ideal for the purpose of this lesson. Students work in small groups (4-5 students) to explore the Pythagorean Theorem using a variety of materials. Depending on the level of your students, you may choose to focus on one of the investigations, add or omit some of the investigations, scaffold them differently, and/or do the investigations simultaneously (such that each group is responsible for one of the investigations and shares their finding with the rest of the class) or in series (such that each group gets to explore all of the investigations in a sequence). 

Materials

Investigation 1

  • Triangle cut-out from grid chart paper
    • base: 4 square units, height: 3 square units
    • label the legs of the triangle as “a” and “b”, and the hypotenuse as “c”
  • Plastic square tiles (3 different colours; 25 of each)
  • Grid chart paper (to be used as the activity board)
  • Instruction card: “Using the tiles and the triangle provided, prove that a2 + b2 = c2.”

Investigation 2

  • 3 sets of tangrams with different colours
  • Grid chart paper (to be used as the activity board)
  • Instruction card: “Use the tangram pieces to show that a2 + b2 = c2. Start by using the smallest triangle. Then try with the medium, then with the large triangle. keep a record of your solutions by tracing the final shapes. *Hint: You may not need all pieces for every solution, but you will need to combine some pieces from all 3 sets of tangrams.”  

Investigation 3

  • 4 congruent triangles (base: 4 inches, height: 3 inches) 
  • Plastic square tiles
  • Instruction card: Manipulate the 4 triangles in the square provided to show that a2 + b2 = c2.”

Investigation 4

  • Pieces of wood: 5, 10, 13, and 26 inches long, labeled with respective lengths
  • Grid chart paper (to be used as the activity board)
  • Calculator
  • Triangular rules
  • Instruction card: “you have to build three slides: 5, 10, and 13 inches long for the Flea Circus. Both height and the base of the slide must be whole number lengths. By leaning the slides provided against a vertical surface, measure and record how high your slides can be and how far the end of the slide is to the base of the ladder.”

Investigations

Investigation 1

Students will explore the Pythagorean Theorem by arranging tiles to show that the total number of tiles needed to cover the squares of the legs (sides “a” and “b”) equal to the number of tiles needed to form the square of the hypotenuse (side “c”).

As a scaffold, you can demonstrate how to make the first square against one of the legs. You can also encourage the students to superimpose a2 and b2 onto c2.  

Investigation 2

Students will explore the Pythagorean Theorem with Tangram pieces.

You may wish to scaffold this investigation as such:

  • Step 1: Place one of the smallest Tangram triangles in the centre of your paper and take around it. Label the longest side of the triangle “c” (hypotenuse) and the other two sides “a” and “b” (legs).
  • Step 2: Use two other smallest Tangram triangles to form a square. Place the squares beside side “a” so that one of the sides of the square and the side “a” touch. Repeat for side “b”. Trace all the pieces on the paper.
  • Step 3: Remove the two squares (each made up of 2 triangles), only leaving the original triangle. Now, use those four pieces to make one big square. The sides of the square should match the side “c” of the triangle. Place the square beside side “c” so that one of the sides of the square and the side “c” touch. Trace all the pieces on the paper.
  • Step 4: Now try this process with the medium Tangram triangle, and the largest Tangram triangle. You may want to use different combination of Tangram pieces. There are many different possibilities. Keep a record by tracing the pieces on the paper.

Investigation 3

Students will explore the Pythagorean Theorem with four identical right angle triangles.

You may wish to scaffold this investigation as such:

  • Step 1: you are given four identical triangles with a base length of 4 inches, an height of 3 inches. Place the four triangles in such a way that you form a square with side lengths of 7 inches, with a small square hole (slanted) in the middle. Trace the square on the paper. (Alternatively, you can draw the square ahead of time and give it to the students). 
  • Step 2: Fill the square hole with plastic square tiles. How many did you need in all? What does that tell you about the length of the hypotenuse of each of the triangles?
  • Step 3: Staying inside the large square drawn, rearrange the triangles to form two square holes that, when combined, can be covered by the same number of square tiles as the original square hole. 

Investigation 4

Students will investigate to find some Pythagorean Triples (a2 + b2 = c2, where a, b, and c are positive integers). This serves as an extension investigation, and could be used as a lead into the introduction of radicals for solving non-Pythagorean triple cases.

You may wish to scaffold this investigation as such:

  • Step 1: Place the 5 inch slide against the wall to create a very steep slide. Measure the height of the slide and the length of the base. Are they whole numbers? Record the measurements.
  • Step 2: if the numbers weren’t whole numbers, move the slide slightly to decrease the steepness of the slide. Measure the new height of the slide and the length of the base. Are they whole numbers now? Record the measurements.
  • Step 3: Repeat he process until both measurements are whole-numbers.
  • Step 4: Now try wit the 10 inch slide. First, make a prediction about the height of the slide and the length of the base. Then repeat steps 1-3 with this new slide. Do you see any relationship between the measurements of 5 inch slide and 10 inch slide?
  • Step 5: Now try with the 13, and 26 inch slides. Do you see any relationship?

References

Investigation 1
http://www.youtube.com/watch?v=jizQ-Ww7jik

Investigation 2
http://aaronburhoe.wordpress.com/2010/07/12/burhoe-6-a-1-the-pythagorean-theorem-with-tangrams/

Investigation 3
http://www.mathopenref.com/pythagorasproof.html

Investigation 4
http://realteachingmeansreallearning.blogspot.ca/2011/02/discovering-pythagorean-theorem.html

Pythagorean Theorem Water Demo
https://www.youtube.com/watch?v=CAkMUdeB06o