In this lesson, students are creating their own building using their developing knowledge of two-dimensional shapes and the formation of 3 dimensional solids from constituent parts. Additionally, they are integrating ideas about aesthetic and functional reasons for architectural choices that they explored in the lessons on Great Buildings of the World.
This is a very sophisticated task, both cognitively and on a purely practical level. Students need demonstration on some specific tasks. Some students need a moderate or significant amount of assistance initially with the manual tasks. I reviewed/ taught the following:
- How to make hinges with card stock to join the sides.
- How to reinforce the hinges with bent pipe cleaners for especially heavy walls.
- How to line up the bottoms of the sides before attaching hinges.
- How to hold down one side as a template to trace the opposite matching side.
- How to measure even rectangles in inches or centimeters by drawing marker lines.
The complexity of the roof is directly tied to the polygons the student chooses to use on the irregular side of their building. Some children may need to make a simple roof, but those who are doing it because their stated purpose is to find the easiest solution need to be pushed further.
Children who make a triangular prism roof are taking this task a step further, so I leave them to this challenge. I provide 1-1 support for children whose choice was a roof with two complex polygonal sides.
In this mini-lesson, I model how to take on a complex task such as roof construction by breaking it into a sequence of rectangular faces.
The students are already in very different stages in their work on this 2nd day of the activity. Some students are meticulously gluing the shapes that constitute their sides (especially the non-rectangular pair) onto the cardstock. Others are already attaching the sides with hinges. A few students cut out overlay shapes to put on one of their sides to represent further 3 dimensional shapes. (Insert photo of Aquiel’sl library w/Greco-Roman columns).
Throughout the lesson I am walking around the room to facilitate productive struggle and stave off frustration. While I think you will find that the students are highly engaged in this task as it addresses kinesthetic and visual learning styles, because there aren’t any restriction on the non-rectangular sides other than that they must be polygons, students can back themselves into a quite complicated situation.
While I am there to assist their thinking, this is a chance to support individual creative tastes rather than imposing an adult's vision. Rather than pointing out the impracticality of some structures, I work to discern the student’s intent and support it. (Insert video of Roman’s White House or Zahory’s hexagon parallelogram sides w/open spaces).
Make sure that throughout the task students can identify basic polygons, and are using their vocabulary to refer to edges, vertices, sides/faces. It is important that they are starting to express specific ideas about how polygons combine to form varied 3 dimensional solids. Modeling with mathematics (MP4) combined with precise language (MP6) will help students contextualize these core geometrical concepts and vocabulary words in a meaningful way.
This child's tenacious struggle to create the building he wanted is the kind of moment that makes me deeply glad I am a teacher. When I pointed out to him at the start of his construction that his irregular sides might make it difficult to put on a roof, he shrugged his shoulders, smiled, and said this is what he wanted to do! Who can argue with that? In fact, as we progressed through the several days of this activity, he was often used as an exemplar, in kid to kid talk, of determination and creativity. The students who made cube shaped buildings finished first, but speed was not a measure of success in this activity, and they quickly set to work on a second building or as architectural consultants to their peers.
To be sure, a divergent, creative thinker like this child does not make things easy for themselves, or their teacher, but nobody is ever bored! This is an activity that certainly would benefit from the support of parent helpers trained to put on rectangular piece roofs. I found that the children who took home their complicated buildings, with an irregular side composed of adjacent hexagons, for example, came back with a flat roof that didn't match the side. While this is a justifiable architectural choice, it dilutes the underlying purpose of building the roof: to measure the perimeter of rectangles that will create a 3d polygonal roof, and to view the beautiful complexity of the 3 dimensional roof itself.
So back to this child, it took a few days of support to get his roof on. Not only did each rectangle need to be measured, but his sides had not been precisely cut, so the rectangular pieces didn't completely line up. Then the paper pieces needed to be attached with a variety of supports that had to be placed by reaching up on to the underside of the roof. This boy doesn't, necessarily talk a lot, and by interacting with him on a self-selected complex task over the course of several days, I was able to engage in meaningful, timely, relevant mathematical discourse that I believe will add to both his written ability to express mathematical concepts and his conceptual understanding of basic geometry.
I randomly place students in partnerships, and have them take turns explaining what they've done so far, with the expectation that they reference the polygons and 3 dimensional solids in their building. At the conclusion of each student's share, their random partner is responsible for making one specific comment (even a restatement) and asking one thoughtful question specifically about mathematical elements.
I've provided an example here of what that sounds like. For example: Did you realize that making a roof with hexagons would mean using rectangles for the roof itself? Not: Why do you have pink paper on this side and pink and green on the opposite side.
We have practiced math discourse for several months now, and this expectation, while rigorous, is fair for most of them. If your students are not experienced in listening and framing their thinking academically, it will likely be difficult to formulate thoughtful questions or comments. Providing an example is a good starting point. Offering students a comment and asking them to judge whether it is appropriate or not is also a helpful practice. Having some students who are comfortable with this framework share first, so that others have time to use that context to process their thinking is another supportive practice. But if/when needed, you can provide a sentence starter. These are two very basic examples:
Comment: In designing your __________ building, you used ______________ to design the ____________.
Question: What would happen if ____________? Did you _____________ ?