G-GMD.B Visualize relationships between two-dimensional and three-dimensional objects
4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
This lesson is another reason that I like CCSS-M. We weren’t talking much about cross sections of 3-D objects before we started teaching CCSS Geometry, and we definitely weren’t determining the 3-D objects formed by rotating 2-D objects.
When I first started looking for ideas for this lesson (it’s not just that we weren’t teaching this topic before, our textbook had very few tasks to go with this topic), I found a Laying the Foundation lesson on Volumes of Revolution. Laying the Foundation is part of the National Math + Science Initiative. Upon further investigation, it looks like they have posted a newer lesson on Using Linear Equations to Define Geometry Solids. We will use it to update our lesson over the summer.
We started by thinking about the solid formed when rotating the rectangle around a line of symmetry.
What would be the volume of the solid?
I sent a Quick Poll. The results provided us an opportunity to attend to precision. Which of these should we mark correct? Why?
How many times have you had a student cube or square the area or volume instead of only the units on a summative assessment? Formative assessment helps us change and improve our practice. Before I had “clickers”, I would have mentioned to my students to be sure they were only cubing the units … I’m not sure that O.F. would have heard me. For O.F. to see her answer compared to the others in the class makes a difference. Her misconception has been corrected, and the rest of us had the opportunity to learn from her mistake.
N.K. also had the opportunity to learn to attend to precision, paying closer attention to whether he is calculating area or volume and using the appropriate units.
The lesson continued, thinking next about rotating a rectangle about the line containing one of its sides (here, the y-axis and then the x-axis), calculating volume, and correcting misconceptions.
And then we moved to creating the shape to rotate in the coordinate plane.
And so the journey continues …