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This past week I had the opportunity to share with a graduate class the Common Core progression of geometric measurement concepts across the grade levels from Kindergarten through eighth grade.

We packed the space between two points of a line with one-inch sticks, and we packed a rectangle with squares, and then we packed a rectangular solid box with cubes. This is the progression of geometric measurement from Kindergarten through fifth grade. The progression is from length measure, which is one-dimensional, to area measure, which is two-dimensional, to volume measure, which is three-dimensional.

The progression continues in sixth grade where students explore the volume of rectangular solids, which have edges with lengths that are fractions. In seventh grade students encounter volume of prisms and pyramids, and in eighth grade volume of cylinders, cones, and spheres.

We noted that beginning with the sixth grade, none of these shapes can be packed using only whole cubes. We talked about the fact that just when students begin to encounter shapes that require cutting up some of the cubes in different ways in order to fill all the gaps, we quite providing students with objects to look at and to handle.

We continued to talk about this as I showed them the objects you can see in the following video. We talked about the value of students seeing and handling these shapes packed with cubes as they are learning about the formulas for finding these volumes.

How would the student’s thinking about the volume of a cylinder be influenced if they could see and handle layers of cubes, where some cubes were cut so that each layer of cubes completely filled that layer with no gaps?

It turned out that three of the students in the class who are sixth grade teachers were in the middle of or just beginning a unit on volume of rectangular solids with fractional edges. After class all three came up to ask if they could borrow some of the rectangular solids that had fractional edges. I believe I had a half dozen of them at the time; you will notice that there are no such solids in the video now–they are all out in classrooms.

While I was excited to see their interest in showing the solids with fractional edges to their students, I had to wonder about how objects like these could be made available for more students. I can do it for myself and for my students because I have the interest, ability, and tools to build these objects, but what about those teachers who are unable to do so?

One of the things I’ve been thinking about is the possibility of 3-D virtual environments that might make it possible for students to have some of the experiences that the concrete objects can provide. What are your thoughts?

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