Drawing Triangles with Circles
I was going back through my materials from the first semester and decided this was something I wanted to share and reflect on a little. This is my attempt at contributing to the ongoing Math Blogotwittersphere Conference. So here goes.
This post is meant to catalog how I used constructions as a way to build intuition about triangles. Because it took a while to fix so many algebra and arithmetic holes, it wasn’t until the semester was almost over and we were beginning triangles that I finally found a spot to get their hands dirty and start constructions. It worked out rather well and I discovered (to my surprise) that teaching the Triangle Inequality is one of my favorite subjects to cover in geometry. Part of that has to do with the way constructions tie together with drawing triangles.
Small disgression: here’s something I’ve realized about how I like to teach: if I can, I would much rather give my students an intuition rather than an explanation. I’ve thought about intuition before and it’s really been the basis for the questions I’ve had about teaching algebra and the number line – a lot of my students struggle with algebra and arithmetic because they don’t have ant intuition to guide them – no underlying foundation to fall back on. I’ve also thought about this in a conversation I had with a colleague comparing our journey’s in becoming educators – he started as a student in the science-education program at my university, while I started as a student in the math-education program at my university. In talking about our experiences, one thing I found fascinating was that one of the bigger emphasis in the science program was on intuition – to paraphrase him: “When we find a student who has the wrong intuition about something, it is out job to mercilessly destroy that intuition without prejudice. If we don’t, that student will always fall back on that intuition when they’re not sure about something, which means they will always be frustrated every time they don’t know something” (personal bias: I like the imagery of ‘mercilessly destroying intuition without prejudice’). Final thought on this – Frank Noschese said it in a way that really stuck out to me: “Teaching is really about creating experiences that allow students to construct meaning.” (Context Disclosure: This quote came from this blog post, where Frank is criticizing the notion that good teaching is really good explanation).
So – my goal was to teach triangles in such a way that my students developed an intuition rather than an explanation. And the way to do this, maybe surprisingly, is with circles.
Essential Question: What does a triangle with sides 5 cm, 7 cm, and 11 cm really look like? Can you accurately visualize it – picture it in your head? Will it be acute? Obtuse? Which angle will be the largest? The smallest? If I gave you toothpicks with that size, how would you construct that triangle? What if I told you to draw it rather than build it – how many times would you have to erase your lines and start again? Is there a better way?
Did you know you can construct a triangle using the following procedure? (1) Draw a segment that is the length of your first side; (2) Using one endpoint as your center, draw a circle that has a radius of your second side; (3) Using the other endpoint, draw a circle that is the radius of your third side, (4) Your circles from (2) and (3) should intersect – draw two segments connecting this point of intersection to the two endpoints of your original segment; (5) viola: you have an accurate triangle. For all of this in worksheet format, see below:
I let my students practice this a bit (this is also ‘learn how to use a compass’ practice) and, in doing so, developed their ability to visualize triangles, segments, and angles. Once my students practiced for a bit, I gave each group 4 triangles to draw with the caveat that one of those triangles was ‘impossible’ (although I didn’t explain what this meant yet). I had them draw their triangles and, in each group, one student had a picture that looked like this:
The moral (which my students described to me, rather than me describing to them): Each triangle had one side that was just way too big. This meant the two circles would never be big enough to intersect, so it’s impossible to draw the triangle. I then gave each student an impossible triangle to draw just to get more practice, but I made sure each group had a triangle where the two smaller sides added up to exactly the length of the third side (like 2, 3, 5 or 4, 4, 8). These constructions are interesting because the two circles do actually intersect, but at exactly one point – in other words, the two circles are tangent to each other (which is a nice way to introduce this piece of vocabulary). However, we all agree: they still don’t make a triangle because there’s no ‘space’ that is enclosed by the three segments – they all just overlap on each other, which isn’t allowed.
By now my students are getting the hang of what makes something possible vs impossible. So I have them make up their own and test it using this applet: http://www.geogebratube.org/student/m3038 (This is also a good applet to see what I’ve been talking about with the circles and segments. If you’ve been having trouble visualizing all of this, play around with the applet for a bit).
Now my students are almost ready to come up with a general rule: that the two smaller sides have to add up to more than the larger side. But again, here’s why I like this: if I were to ask them ‘why?’, they can say ‘because when you’re constructing this triangle, the two circles won’t intersect, so it’s impossible to make the triangle’. And if a student can say this to me, I’m a lot more confident that they’re able to visualize what’s actually happening with the Triangle Inequality theorem (which I called the Impossible Triangle Theorem) and that they’ll not only remember it, but maybe even use it intelligently one day.
Using this intuition, you can also more easily explain why the largest angle is always opposite the longest side (although, in my opinion, this isn’t too hard of a sell). Y0u can also explain why Angle-Side-Side is not a valid way to prove two triangles are congruent – you can explore this using this applet: http://www.geogebratube.org/student/m3040.
So, this is something I’m saving for next year – I feel like I managed to impart a sense of intuition which carried my students a long way, and they ‘discovered’ something using my favorite method of all: playing around and trying things out until you notice a pattern, then verifying the pattern and internalizing it. Actually, if I were to do this again next year, I might even do this in the very beginning of the year when I’m trying to get my students used to ‘visualizing’ what their shapes look like – I feel like most books and curriculum maps put this in with centers of triangles or even triangle congruency theorems (like SSS or SAS), but it’s really an exercise in exploring how to construct accurate triangles and the limitations of those constructions (which is really all that the triangle inequality theorem is). So maybe this belongs in a unit on constructions rather than a unit on triangles. I dunno – these last sentences spilled over into the realm of ‘sequencing a geometry curriclum’ musings.