## November 10, 2013

### Properties of Triangles

Last year when we started talking about triangles one of my colleagues did the relationship between angles and sides first (the largest angle is opposite the longest side etc.) and found great success.  The angle properties of isosceles and equilateral triangles easily follow the definition, the hypotenuse is obviously the longest side of a right triangle, the impossibility of an obtuse equilateral triangle is quickly apparent. So I wrote down that I would do that first this year. But then I forgot the definition of first, I had students define types of triangles and try to draw obtuse equilateral triangles before discovering this relationship. I know now, first means first. At the very beginning. As soon as we say the word triangle, we should be doing the exploration relating sides to angles.

A fun way to transition from lines to triangles is this activity from Mr. Stadel. I retyped it to get the instructions and diagram on one page, it's exactly the same as his though.

With just a single extra line, you can prove that the angles of a triangle add up to 180:

 http://en.wikibooks.org/wiki/Trigonometry/Proof:_Angles_sum_to_180

Now we've mentioned the word triangle, time to explore sides and angles! This sheet is adapted from one in my textbook. I edited it with my colleagues. It was awesome. Kids sat quietly cutting out their side lengths, trying triangles and figuring out what worked. They weren't just quiet because they were cutting and tracing, there was thinking involved in the process. They were noticing and wondering, because, as one student declared, "You always ask us that!" She was complaining that I asked her to write something down when she finished, but what a great complaint 'You always want to know what we think!' I smiled, I have no idea what thoughts ran through her head, but then she commenced recording her ideas.

I'd have a picture of graph paper rulers, but I forgot to take one. Might remember tomorrow. Idea: cut out rectangles 1 box wide in the length needed. Compasses, wooden rulers and plastic rulers all work as well, but compasses are tough to hold steady and most rulers are long and thus unwieldy. We used graph paper just a bit larger than the standard size. Give kids one full sheet to draw on and a quarter sheet to cut from and they're good to go.

Goals of this activity: figure out what side lengths make a triangle (triangle inequality) and discover the angle-side relationship. Having the side lengths in decreasing order would make the second discovery more obvious. I chose to have kids work for it a bit, but am open to change. This page was typeset in LaTeX, if you want the original file so you can edit it please let me know.

My next step would have been to define all the words that we use to classify triangles, but since I already did that it will be to have kids work through stations practicing and applying the rules they discovered. I'm unreasonably excited about the coupon holder I got at Staples for keeping all of my station activities in order. Organizers are such fun!