Pythagorean theorem right triangles tasks

Garbage Can Task

The following task happened by accident:

I was about to introduce a problem to my Math 9 Enriched class that we were going to complete with group whiteboards. Before I could introduce, life got in the way. Students wanted to know about their most recent examination. As I launched into a speech on their performance, a student got up to sharpen their pencil. She walked right in front of me. I made a comment, and she replied that the garbage can should be in the back corner where it would be more convenient. 

Pythagorean theorem tasks

Road Building Task

The Pythagorean Theorem is often taught in isolation. It has connections to solving equations, but often appears in curriculum long before other equations involving radicals. It also has unique ties to both radicals as well as geometry.

Despite these connections, the theorem has developed the reputation of a surface skill. It involves the  repetition of the rule alongside numerous iterations. Something so fundamental to geometry is reduced to a droning chorus of:

” ‘a’ squared plus ‘b’ squared equals ‘c’ squared “
area circles Pythagorean theorem right triangles tasks

Sprinkler Task

I am frustratingly mathematical. Ask my wife. I see the world as a combination of, in the words of David Berlinski, absolutely elementary mathematics.(AEM). The path of a yo-yo, the tiles in the mall, and the trail of wetness after a bike rides through a puddle are all dissected with simple, mathematical phenomenon. The nice part about AEM is that I can talk about it to almost anyone. People are (vaguely) familiar with graphs, geometric patterns, and circles even if they can’t decipher what practical implications they have on their city block. Unfortunately, people (and students) don’t often want to hear about them–they need to see them.

I can remember the look on my mother’s face when I broke out the silverware to show her that the restaurant table corner was not square. Without a ruler, I showed her that trigonometry allows us to rely on ratio rather than set measurements. As I was in the midst of showing her that the 3-4-5 knife-length rule was breached, the waitress came. Mom was horrified; I was thrilled.

proofs Pythagorean theorem similar triangles

Pythagorean Triples Part 2: Teacher Learning

If you are not careful, teaching can become very boring, very quickly. Most teachers of specialized areas teach the same content arranged in the same manner numerous times throughout a career. It is no wonder teachers are constantly warned of burnout. Opening up space for student initiative serves a two-fold purpose:

First, the extra freedom allows students to create significance in memorable ways.
Second, the sheer variety of student queries can raise questions for teachers.

investigation Pythagorean theorem

Pythagorean Triples Part 1: Student Strategies

The school year is winding down for me and my project-based grade ten classes. I have found myself looking at the curriculum more and more as the final day approaches. I was told by many that content coverage would be impossible in a project-based setting; this only made me more anxious. Compounding this problem, I needed a substitute teacher for a day and do not like throwing them into a project setting without any briefing. In order to accommodate them, I chose to photocopy a worksheet on the Pythagorean Theorem for my students while I was gone. When I alerted them of this, the response was clear:

PBL Pythagorean theorem similar triangles tasks trigonometry

Trigonometric Mini Golf

Christmas time brings immense stress for math teachers, at least in my division and province. As the days dwindle away, teachers begin to get a more accurate picture of how much they must cover before semester’s end. Once again, I found myself in this position with my Grade 10 Foundations and Pre-calculus class. (Saskatchewan Curriculum) My original plans called for 20 teaching days to adequately cover, in my opinion, the topics of trigonometry and systems of linear equations. Of course, by the time I sat down to calculate this I only had 11 remaining.
In previous years I would have panicked and switched into jam-packed lectures to “cover” all the content. This year I decided to re-think that approach. I wanted to find a project or anchor activity that could facilitate a wide swath of outcomes and motivate a high level of learning so close to holidays. I tried several creations, but settled on this one for its native curiosity and deep flexibility.
Pythagorean theorem reflection

Life Without Euclid

This post has nothing to do with geometry. I guess I can’t say that exactly (because of the possible geometric representations), but I am not dealing directly with these. I am always intrigued when I think like I want my students to think. It is these moments that keep me going into the classroom hoping for new understandings. There have been times this year where students have made connections that I never have. These innocent realizations are mathematics manifested in its purest form. A similar experience happened to me this morning.