Tapping It Up a Notch: Pythagorean Theorem – Part 2
In our last post, we used an inquiry/discovery approach to help students visualize how we can find the hypotenuse of a right-angle triangle when given the lengths of the two legs.
In this post, we will now introduce the General Case for Pythagorean Theorem in an attempt to use the same visual model to derive the formula for Pythagorean Theorem.
Using Pythagorean Theorem to Find the Length of the Hypotenuse
Visual Representation of Any Right Triangle [General Case]
Summary of Pythagorean Theorem Video
Starting With The General Case Visually
We now start with variables representing any side lengths for a right-angle triangle.
Connect Squaring Side-Lengths to Area
Once again, we show that squaring side lengths on the right-angle triangle will yield an area since we are multiplying length by width (or side by side, in this case):
Segment the Area of the Shortest Leg into Pieces
In order to preserve the visualization of the sum of the squares of both legs being equivalent to the square of the length of the hypotenuse, we will need to chop up one of the areas into pieces:
Show The Sum of the Squares of the Leg Lengths Algebraically
As we did in the 3, 4, 5 case in the previous post, we now must use algebra to show what the sum of the squares of the leg lengths look like. This can help students start to see what the formula must be.
Finding the Length of the Hypotenuse
In the previous case, finding the side length was pretty easy for students to do without necessarily consciously thinking about opposite operations. In this case, we must square-root the resulting area to find the length of the hypotenuse.
Introduce c^2 as a Variable to Replace a^2 + b^2
Using substitution, we will replace a^2 + b^2 with a new variable, c^2.
After students are comfortable with the algebra behind Pythagorean Theorem, you may choose to discuss this more deeply. Having students understand that we are actually substituting c with the square-root of a^2 + b^2 could be beneficial as they get closer to advanced functions and calculus.
Deriving the Formula for Pythagorean Theorem
Students may now be more comfortable understanding and using the algebraic representation of the Pythagorean Theorem Formula:
Another post is forthcoming to help students better connect the visual representation of Pythagorean Theorem and the algebraic representation.
What do you think? How can we improve the introduction of this very important mathematical concept? Leave a comment below!
Other Related Pythagorean Theorem Posts:
Tapping It Up a Notch: Pythagorean Theorem - Part 1 Over the past school year, I have been making attempts to create resources that allow students to better visualize math, build spatial reasoning skills and make connections to the algebraic representation. While some hardcore mathletes might balk at much of these attempts as not being real mathematical proofs, my intention is to help stud...
Tapping It Up a Notch: Pythagorean Theorem - Part 3 In the first part, we used an inquiry/discovery approach to help students visualize how Pythagorean Theorem works, then followed that with a visualization of the general proof / derivation of the formula to show how we can find the hypotenuse of a right-angle triangle when given the lengths of the two legs. In this post, we will now ma...
Who will reach the taco cart first?
Understanding and Applying the Pythagorean Theorem The Taco Cart is another great 3 Act Math Task by Dan Meyer that asks the perplexing question of which path should each person choose to get to a taco food cart just up the road. While most students could quickly identify the shorter route using basic logic, Meyer throws in a curve-ball when one path forces one person to walk in the sand (...