A few months ago, I got an email from one of the art teachers in the school. She was offering her classroom, a makerspace of sorts, and her creative expertise to any teacher that wanted to bring their class down.

I jumped at the opportunity. After all, what have I been enjoying most about math lately? Making and appreciating #mathart.

So. We brainstormed and ultimately settled on doing a project that would lead into our upcoming similarity unit. It would have the kids constructing equilateral triangles descending in size, alternating colors, each triangle with a side length the same as the height of the previous triangle.

The students had never done constructions before, so it was a good introduction to using a compass, and it was also a chance for the students to play with the artistic concepts of structure, contrasting color choice, and composition. The kids had fun, and the results were stunning!

This whole project took about two 50 minute class periods. Toward the end of the second day, one of the students said: “Mrs. von Oy. This is fun and all, but it doesn’t seem like we’re really doing much math.”

Well. Turns out we are. Lots of it.

And also? Maybe math just **is** fun, and we teachers need to bring the fun out a little more often.

So here’s the plan for the “triangle art” debrief discussion:

**Why did that particular compass construction create an equilateral triangle?**(and did it matter if your compass was too loose to keep its radius the same?)**Did it matter what size triangle you started with?****How many triangles could you make if you were to keep going?**(Most students made about 10 triangles. This is a great time to begin getting kids to think about asymptotic behavior, even to think about limits)**What is the difference between how many triangles you could make out of paper, and how many you could imagine mathematically?**(a question to provoke conversation about the nature of mathematical abstraction. Perhaps I will have students read this blog post by Ben Orlin in class or for homework: There’s No Such Thing as Triangles)**If you made the spiral pattern (like the one below), could you ever get all the way around, and how do you know?**(A student conjectured today that it would be impossible because the triangles were getting smaller. That led to a great discussion of side lengths getting shorter vs. angles getting smaller…or do they stay the same measure? Would love to have the whole class in on this conversation)**If the side length of the first triangle was 10 cm, what would the side length of the second triangle be?**(Let the students struggle with this one. They know the Pythagorean Theorem, but haven’t spent much time using it for interesting problems yet. I want to see them figure this out on their own.)

**What are some questions you would ask your students? Send some comments my way…I’d love to facilitate an even more robust discussion.**

One of the things I’ve been thinking about lately is the square. Not having had a clue about Euclid and having no grasp of the idea of generalization in math, it’s no wonder that shapes and math didn’t seem to exist in the same universe. Once the idea that a square is considered a generalized unit in the coordinate plane, shapes started to make sense. After that, the next step was to see a circle as a generalization for a cycle. Seeing you all doing your desmos thing, and seeing this math/art gets me pondering over the generalizations that are embedded in all of these shapes. The spiral describes the natural log, growth and decay? Am loving seeing the visuals you are creating here and hoping others will write out the way they make connections between math and shapes. Such great work!

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beautiful!

you could also ask the students to calculate the area of the first triangle (always good practice), the second triangle, the third triangle, and then ask them to predict the area of the 15th triangle, justify their prediction, tell them that a classmate started with a triangle whose area was 12 cm^2, and that the side of the 3rd triangle was 3x smaller than the first and ask what the area of the 3rd triangle is.

another thing i’m just wondering is how would some of these look if the students also drew the circumcircle for each triangle?

finally an aesthetics question: would this have worked as well with isosceles triangles? or even scalene triangles? what might change if you tiled and overlapped those?

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All great questions! Thank you!!

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Love the art integration here. The student work turned out phenomenal. The only similar (no pun intended) activity I have done is the Wheel or Spiral of Theodorus… it really helps students see what happens when you square an irrational number. It’s also interesting to see how students decorate it based on what they see. Here are the examples http://joyceh1.blogspot.com/2016/05/day-157-volume-of-cone-pyramid-penny.html This is an annual project I will continue to keep doing.

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Amazing ideas! Can I ask what grade this was?

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Hi Jackie, I did this with mostly 10th grade students. I imagine this project could be a great entry point for taking about math on many different levels and with many different ages of students.

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I didn’t see any directions on how you have the students make the triangles using the compass. Did I miss that?

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Hi Jessica, we used the standard construction for an equilateral triangle. If you’re not familiar, here’s a video explaining how it’s done: https://www.youtube.com/watch?v=t-ZtoNhEYWQ

To find the height of each triangle (to use as the sidelength of the next one), the easiest way is to cut the triangle out and fold it in half. You can also construct the midpoint.

Each subsequent triangle gets constructed with the “arms” of the compass stretched to be exactly the height of the previous one.

Our triangles had some variation because our compasses were getting loose and didn’t hold their settings very well. But we did our best and had fun! 🙂

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I am looking forward to trying out the sequence of constructions and when or if it spirals around onto itself I am thinking about the kind of tiling design I could make with a complete revolution. I am really interested in having tasks like this in my resource bag, especially as it requires students to use and apply Pythagoras without it being ‘obvious’. This is a good way tob’tedt’ out whether students really do have Pythagoras in their problem solving toolkits. Thank you

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