Four Projects for Special Right Triangles
Over the weekend, I had several strokes of insight and put together 4 origami-based projects for my students. Here they are in their raw form.
Content Covered: Finding missing side lengths of isosceles right triangles (45-45-90) and equilateral triangles (which are composed of two 30-60-90 triangles). Also using these lengths to find areas of squares and equilateral triangles. Some projects involve basic knowledge of volume and surface area.
Practice Problems: First better make sure they know how to do the basics…
Quick Note: for now, I’m purposely avoiding classifying these by their angles (ie: 30-60-90), which is why the second worksheet is all about equilateral triangles. The way I’m teaching this unit, this section is all about finding patterns, so I’ve been emphasizing the patterns related to equilateral triangles rather than focusing on the technicalities of the 30-60-90 relationship. However, in a week when I get to trig ratios, we’ll revisit these and make a big deal about the angles. That’s my plan, anyway.
Project #1: The Conch Shell. Focus on isosceles right triangles and secretly get them thinking about infinite series…
Origami Instructions (Found these online somewhere a long time ago, so not sure who to credit this to. Sorry…)
Project #2: Volume of a Square Box. Focus on isosceles right triangles as well and secretly about how all squares are similar to each other
Origami Instructions (Actually, this is a link to an article from Mathematics Teaching in the Middle School published by NCTM. The instructions are embedded in the article along with several strategies students used to solve the problem)
Project #3: Dollar Tetrahedron. Focus on equilateral triangles and secretly about conspiracy theories in the US Dollar. Be sure to read my full disclosure note below.
Full Disclosure: This project is a retooled version of how Kate Nowak teaches 30-60-90 right triangles – see her blog post here: http://function-of-time.blogspot.com/2010/11/special-right-triangles.html. I stole all the images of the dollars from her website – I hope she doesn’t mind. I recommend reading her blog post if you’re looking at this project
Project #4: Origami Octahedron. Focus on equilateral triangles and secretly about cool shortest path problems.
Personal Teaching Note: With the way things went this year, all the content from now to the end of the year is post high-stakes-big-deal-standardized-test (they took the big Arizona exit exam a week ago). This has substantially changed the culture of my classroom in two ways. The pessimist in me is scared out of my mind that my students will stop caring and the next 4 weeks will be hell – I can no longer motivate some of the more boring and rote procedures by simply saying ‘You don’t have to like it, but you have to know it if you want to graduate’. The optimist in me, however, sees this is a bit freeing – I no longer have some state exit exam holding me down to the level of boring and rote procedures that I feel like I need to teach my students. I can go outside the box if I want to.
These projects are me trying to be the optimist – trying to challenge my students with less structure than they’re used to and invite them to be more free with their problem solving. There are many steps in between the question and the answer, but also many ways for them to investigate the answer. I’m not sure how they’ll respond – this could be wonderful or it could be hellish. I’m waiting to see how the next few days go.
Update 4/19: I just spent the day with my students letting them wrestle through these and I want to write down my thoughts while they’re still fresh in my head:
- The Square Box paper needs some logistical retuning. The directions are unclear and confusing – they require students to pause partway through making the box to examine something. It’s also not clear when they should begin to follow the separate origami instructions and when they should be working problems on the worksheet. That one needs retuning
- Biggest Roadblock: Students don’t write anything down. Tomorrow I’m telling them to label every single piece of information they know, then just start finding lengths until they hopefully end up with what they were looking for (which is actually a totally legitimate problem solving strategy – just start doing stuff). But, this starts with labeling your paper and writing all over it.
- I thought the octahedron project would be the hardest one to solve and that the conch shell would be the easiest. From my experiences today, it almost seems like it’s the reverse. In fact, both of the equilateral triangle projects are relatively straightforward once you find the entrance to the problem (which you can get to if you draw some pictures).
- Again, the directions need to be retooled so it’s easier to understand when you’re supposed to make the origami shape and when you’re supposed to answer questions
- I could use some ‘basics of good folding’ handouts or posters or something reminding kids that sharp creases are important and that precision matters.
Overall I’m pretty satisfied with how things are going, but I’m not sure I’ve given enough time for everyone to complete. The impression I’m getting from students is: There is a high threshold for figuring out how to start the problem (ie: finding that first measurements, etc), but once you’ve found that door, everything comes together nicely and you feel like you’ve done something meaningful in answering these problems.