Borrowing from the MTBoS

I’m guessing that most of you reading this are familiar with the awkward acronym for the Math Twitter Blog-oSphere – one of the joys of tapping into this community is that they are remarkably generous about sharing ideas and resources. Today in our Geometry classes (I teach only one of the five sections we have at our school) we used an activity written by Kate Nowak (@k8nowak on twitter). It is an activity based in GeoGebra and allows the student to explore the ratio between lengths of legs in a right triangle. You can find the document we used here . I modified (very slightly) the document that Kate originally posted here. Next time I use it I will tweak it a bit. I have only twelve students in my class and chose not to explicitly team them up. They talked with their neighbors as they are usually encouraged to. However, the directions either need to be tweaked so that team references are excluded or I need to clearly team them up. I am also debating question 7. A number of students did not make the explicit leap from using the ratio they found on page 1 and using it here. I don’t necessarily want to give away too much but I may add a little prompt that they should consider the work that they have already completed. We set up a google spreadsheet and in the next couple of days I will refer to this repeatedly to show that different students working on different triangles were arriving at the same ratio. We make a big explicit deal about scale factors between similar figures. I do not think we spent enough time pointing out that scale factors within figures will also match up for similar figures. I will definitely make this more of a point of emphasis next time through my text.

I cannot thank Kate enough for sharing this activity. My students worked well and I am convinced that they will have a more solid grasp of trig ratios moving forward. As I plan out the rest of the unit I am also going to be borrowing from Sam Shah’s latest post about trig. You can find that over here.

Man – the benefits my students are reaping from people that they will never meet – such as Kate Nowak, Jennifer Silverman,  John Golden, Jed Butler, Sam Shah, Pamela Wilson, Meg Craig, and so many more – is just remarkable.

Show but Don’t Tell

So in Geometry today we began to study the ‘special’ right triangles and I had an idea last night that I wanted to try. I handed each of my students two pieces of paper, a ruler, and a protractor. On the first page I asked them to draw an isosceles right triangle on each side and asked them to have the legs of their triangles be different lengths. I polled the students and had them tell me one of their leg lengths. I then asked them to find the length of the hypotenuse and tell me what number they get when dividing the hypotenuse by the leg length. I, of course, got a variety of answers all of which hovered around 1.4. Some students used the Pythagorean theorem and gave me decimal approximations. Some used the Pythagorean theorem and gave me radical answers. Some measured the hypotenuse with their rulers. I asked them why these answers seemed so close to each other – I specifically avoided the word similar here. Luckily, one of my students told me that all the isosceles right triangles were similar to each other. I pushed back a bit and asked what that had to do with ratios within one triangle. We usually discuss similarity ratios between triangles. The explanations from the students were not as concise as I hoped but we all seemed comfortable that rations within a triangle will be the same when looking at two triangles (or in this case 12) that are similar to each other. Since a few students used radicals we had the exact ratio in front of us and a quick solution using algebra confirmed that the ratio was the square root of 2. Success!

Next up I asked them to draw two equilateral triangles and construct an altitude. Now I asked for the ratio between the altitude and a side length. These answers all hovered around 0.87. We were running out of time now so I did a little more telling than I wanted to but we saw the ratio for the three sides of this new right triangle were 1 : square root of 3 : 2

I have to say I was pleased with their persistence, with their measuring/equation solving, and with the idea that we could see these ratios without simply giving them formulas to try and remember. I may be an incurable optimist, but it feels to me that these ratios will be easier to remember at this point. Now I need to have the discipline to avoid using the words for the trig ratios for at least a few days. I am going to steal ideas from Kate Nowak (here is her trig blog post) and Jennifer Wilson (you can find her trig wisdom here) as I attempt to shepherd my Geometry students through the tangle of right triangle trig. I feel that we had a good start today!


So, Kate Nowak over at has asked folks to consider why they blog. I’ll take a swing at blogging about why I blog. I’ll stick to her questions and try to make sense of this in some way. I don’t know that I have much to say that moves the conversation forward, but I’ll try

1. What hooked you on reading the blogs? Was it a particular post or person? Was it an initiative by the nice MTBoS folks? A colleague in your building got you into it? Desperation?

I’d be reluctant to call it desperation, but when I stopped being a student in 2007 and moved to a new school far from my previous home, I found myself really looking for stimulus. Much of it came from one of my administrators and some of it came from other colleagues. I still had student access to databases and was reading journal articles, but I also started poking around and finding some ideas on the net – I’d like to say it was around 2009 by then. I was home for awhile when my little girl was born and I remember starting to read some blogs I had run across. The first two writers I remember really feeling attached to – in a virtual way – were Dan Meyer and Sam Shah. I can’t swear to the timeline, but this is the past that I remember now.

2. What keeps you coming back? What’s the biggest thing you get out of reading and/or commenting?

I feel a bit restless intellectually and professionally at times. I enjoy the people I work with and I enjoy picking their brains. But, I think that when I was in my doc program I got spoiled by the level of discussion/debate and by the frequency with which new ideas/techniques were flying across my radar. I think that the tingle I get in the mornings when I start opening my email links to the blogs I subscribe to is a way to recapture some of that sensation. It’s rewarding to feel part of such a large community of people anxious to share. I’ve been in the classroom since 1987 as a teacher and I can’t even remember how isolated I must have felt all those years before I had the ability to reach out the way we do now. Again, I had some colleagues I loved, but the sheer amount of interaction that is possible now is just mind-boggling. A little intimidating, in fact.

3. If you write, why do you write? What’s the biggest thing you get out of it?

Is it a little too self serving to say that I get a sense of validation from my writing? That, I’d have to admit, is my first motivation. Organizing my thoughts and trying to make sense of them is the second motivation. The third would be my need to not simply feel like I am not just a parasite. I hope to, in some small measure, add something to the culture,knowledge, and experience that is being shared so freely on the blog world.

4. If you chose to enter a room where I was going to talk about blogging for an hour (or however long you could stand it), what would you hope to be hearing from me? MTBoS cheerleading and/or tourism? How-to’s? Stories?

Some stories of what you see as the arc of this community. Some tips about how to forge and maintain connections. Some vision statement of what you see as the near future of this community endeavor.