My daughter is 11 years old and is currently working hard in 6th grade. She is always curious about learning more mathematics and sometimes becomes the pilot tester for EFFL lessons we are developing. Recently, we were out for a walk, when she asked:
"Wouldn't it be faster if we just cut diagonal across this field?"
I responded by asking her if she thought it would actually be shorter and how much time we might save by taking her shortcut. When we arrived home, I went to work designing some lessons that would help answer these questions. A few hours later and I was testing out the lessons with my daughter and 8-year-old son. After making a mess with broken spaghetti pieces and having some discussion, my daughter made the following realization:
"So a^2 + b^2 = c^2!!! I think I have heard of this one before"
She had discovered the Pythagorean Theorem! This is one of my proudest dad moments to date.
What are the Learning Targets for these lessons?
Learning Targets Day 1
Be able to identify the hypotenuse and legs of a right triangle.
Understand that the hypotenuse is the longest side of a right triangle, but is shorter than the sum of the other two sides.
Learning Targets Day 2
Understand the relationship between the lengths of sides in a right triangle (Pythagorean Theorem).
Learning Targets Day 3
Given the length of two sides of a right triangle, be able to find the length of the unknown side.
The biggest difference you will notice in these lessons as compared to a more traditional Pythagorean Theorem lesson is that students do some thinking and experiencing to arrive at the Pythagorean Theorem, rather than the teacher simply telling students the relationship and then asking them to apply it. This is at the very core of the Experience First, Formalize Later (EFFL) teaching philosophy.
What Level Should These Lessons Be Taught?
The Common Core math standards calls for students to be introduced to the Pythagorean Theorem in 8th grade, but this lesson is low-floor enough that it could be used earlier.
When teaching this to middle school students, it is important that you don't skip over Day 1. Day 1 provides the reason why we might need the Pythagorean Theorem. You could certainly combine Day 1 and Day 2 into one lesson.
Day 2 and 3 could be used in a high school Geometry classroom, but would likely need to be extended further. The extension would likely include examples with non-integer solutions and also a proof of the Pythagorean Theorem.
Feel free to share these lessons with your colleagues who might teach the Pythagorean Theorem. Have them make a reply to this blog post with any feedback or insight that might help other teachers.
Activity: Is the Shortcut Actually Shorter?
Answer Keys: PDF