The Stained Glass Project

Recognize Pi in the red?

I made it a personal goal of mine to focus on projects during my 2nd semester, so here’s my big culminating polygon project for the blogotwittersphere. It’s not perfect, but it’s amenable – go ahead and make it yours.

The Outline: You are proposing a design for a stained glass window for an architect. Your design can only use regular polygons. Due to the nature of the glass, there can be no overlaps or gaps in your design – all of the shapes must fit together perfectly. As part of your proposal, you must include the cost of your design. In order to determine this, the architect has told you that each polygon must have a side length of 16 ft and that glass costs $5 per every 8 square feet of glass. Your Mission: Create a design & write a proposal that includes the cost of your design separated by color.

Additional Materials:

Guidelines & Calculation Chart

Rubric

Polygons to cut out: Hexagons & Dodecagons; Hex, Square, Decagons; Triangles, Octagons, Squares, Triangles

Concepts Covered: This project is the culmination of my entire 4th quarter – it includes trigonometry, area of regular polygons, and some real-world ratio problems which I connect to similarity. By this point in the year, my students have used calculators to solve right triangles (with the exception of inverse trig functions) and they’ve definitely worked with ratios. They’ve also used central and interior angle properties of regular polygons to find the area of a regular polygon, which also requires trig. Sidenote: finding the area of a regular polygon is one of my favorite topics because (1) it’s a long process that requires you to keep track of different measurements and numbers, (2) the numbers have meaning, so you constantly have to ask yourself ‘okay – now that I have this value, why did I need it again?’, and (3) you feel like you’re doing some meaningful work when you finish. The first few days we do this, I go on a high-five rampage as my kids start to get the hang of these – it’s very exciting. Anyway – as the project is now, it’s meant to culminate a unit on area of regular polygons.

Click after the jump to read more reflections and see more results

Some More Results:

     

Eskimo holding the sun

Aliens holding hands on Pluto

The Math Behind It: Is actually really friggin cool! The question I’m secretly having them answer is: what are the possible ways to tile the plane using only regular polygons? One necessary piece of information is that at any one vertex, all of the interior angles must sum to 360 – if the sum is less than 360 then you have a gap, and if the sum is larger than 360 then you have an overlap. One thing I like about this is that there is an answer to this question, which you can look into in detail on Wikipedia: http://en.wikipedia.org/wiki/Regular_tiling. They have connections to super cool polyhedras like the Snub polyhedra and such. Which I don’t talk about in class, but I’m secretly thinking to myself “You’re doing some cool stuff right now and you don’t even know it!”

So, in essence, there are a finite number of patterns you can use to tile the plane, which my students start to discover. Once they have a pattern, they do the typical area calculations to find the areas and then use the ratios to figure out the cost. One thing I like a lot about this project is that if a student colors their shape in such a way that different shapes are the same color (ie: six squares are red and 2 hexagons are red), then they suddenly have to confront the problem of figuring out how to get the area of just the red shapes, which requires keeping track of the areas of individual shapes and multiplying them by the appropriate quantity, etc.

 Implementation: This project served as the entire motivation for my unit on area of regular polygons – I introduced this project at the beginning of the unit and showed them some examples, then told them that in order to create this design, we needed to know about the (1) angles of polygons and the (2) area of polygons. From there, there next few weeks was a typical unit – a day of conceptual, then a few days of practice and nuance – but I tried to tie every lesson back to the project somehow. About a week after I introduced the project, I let them play around with some pieces I had already cut out – this is where I developed the ‘angles must sum to 360’ rule that they need to create their design. I’m still working on the best way to do this. After a week and a half or so, I let students work on this in class for a few days and gave them an exit ticket everyday somehow related to the project. For example, Day 1: sketch what your design will look like; Day 2: calculate the area of  one of the shapes you plan on using in your design; Day 3: tell me the cost of one color in your design.

For Next Year: I definitely need to change my measurements in my project – a 16 ft side length is completely unreasonable, and $5 per 8 square feet led to some outrageous costs. If you can’t tell, I made these numbers up as I was updating this from last year – I’ll need to update them again next year. I think next year I’ll also change up how I implement the project – instead of having 3 solid days at the end of the unit to work through this, I might have 1 day peppered throughout the unit as we cover the material that they need. This is especially true for the area calculations – many of the projects had small area mistakes, which ruined every calculation afterwards. I think I need to weigh the area calculations much higher than everything else so my students realize how important it is, and I should have my students turn in their calculations early so I can check and correct them. I’m still working on fine-tuning my feedback loop, especially with projects.

I also realized that with a few tweaks, this project could tie into other 2nd-semester geometry topics. If I allow students to use special quadrilaterals in their designs (parallelograms, kites, rhombuses, etc), then the space of possible designs gets gigantic. I could also add requirements or questions regarding symmetry: is there rotational symmetry? translational symmetry? reflexive symmetry? I’m slowly realizing that there’s room to turn this into an even broader project than it is now. Which is sort of exciting – what if I had this project going on all year? Or did multiple versions of it?

A Note on Projects in General: Sometime soon, I’m going to decompress my experience this semester in assigning projects. So far, I’ve assigned 4: this one, the Special Right Triangles projects, the Quadrilateral Facebook Project, and an Escher Rotation project (which I never posted about, but may do so soon). I didn’t really know what to expect this year – I was mostly hoping to gather data about what the experience was like, get some exemplar works so I can show them to future years, and see if this process was something reasonable for me to continue doing. So, that’s another reason I’m posting this: eventually I need to figure out how this whole project-focused semester went and how I can make it better next year.