BIG IDEA 1
Connecting 2D and 3D Worlds

Researchers have recently shown that when we work on a mathematics problem, five areas of the brain are involved, and two of them are visual pathways (Menon, 2015) (as I explained in the introduction to this book). Our brains are helped when we work visually; and when we connect visuals with numbers, important brain connections occur. In addition, other brain areas are involved when we touch, move things, and interact physically with mathematical ideas. The area of research concerning physical touch and movement is known as "embodied cognition," and researchers in this field point out the importance of students' holding mathematical ideas in the motor and perceptual areas of the brain (Nemirovsky, Rasmussen, Sweeney, & Wawro, 2012), which comes about when they learn mathematical ideas through touch and movement. In this big idea, students are learning about the features of 3D shapes. This is an area that is particularly important to experience physically, as students will not develop a complete understanding of the mathematical features of shapes if they only see them in two-dimensional pictures in textbooks.

When we taught 83 middle school students a few summers ago, we invited them to build larger cubes out of sugar cubes. A year later, one of the students told me that he still remembers the meaning of "1 cubed" by recalling the feel and the features of the small cube he held in his hands. He told me that it was continuing to help him as he learned geometry. The three activities in this big idea give students a physical opportunity to hold objects in their hands and experience their different features.

In the Visualize activity, students are invited to slice 3D objects to make different two-dimensional shapes. Students will enjoy working with clay, and they will learn about the three-dimensional objects they hold in their hands as they touch and feel the dimensions. As they form the shapes and then sketch them, they will be able to make connections between areas of the brain that deal with physical touch and with drawing.

In the Play activity, students will get the opportunity to explore further with the objects they build out of clay. This activity also includes an element we design into tasks wherever possible-giving students choice. This is something that will increase their interest, which, in turn, will increase their learning and achievement.

Our Investigate activity poses questions that we hope will enable students to struggle, as that causes positive brain activity, and to extend their ideas to high levels. The questions have an openness that is rare in textbooks but that is important, as this openness enables students to develop their own ideas and to develop mathematical thinking in response to ill-defined problems of the type they will meet in the world outside school. We pose the following questions:

• How could you slice this solid so that the face that is made has the same area as the base?
• How could you slice it so that the shape has an area bigger than the base?

Both questions will give students opportunities to think deeply, to wonder about relationships, and to connect ideas.

Jo Boaler

References

1. Menon, V. (2015). Salience network. In A. W. Toga (Ed.), Brain mapping: An encyclopedic reference (Vol. 2, pp. 597-611). San Diego, CA: Academic Press.
2. Nemirovsky, R., Rasmussen, C., Sweeney, G., & Wawro, M. (2012). When the classroom floor becomes the complex plane: Addition and multiplication as ways of bodily navigation. Journal of the Learning Sciences, 21(2), 287-323. doi:10.1080/10508406.2011.611445

Seeing Slices

Snapshot

Students visualize and explore the two-dimensional shapes that can be made by slicing a rectangular solid.

Connection to CCSS

7.G.3

Agenda

• Rectangular solid made of clay
• Cutting tool (dental floss or a wire cutter)

• Clay, enough for each group to form a rectangular solid
• Cutting tool (dental floss or a wire cutter), for each group
• Regular and isometric dot paper (see appendix), multiple sheets per group

To the Teacher

For this activity and the others in this big idea, we want students to have the chance to physically interact with three-dimensional figures and have the opportunity to slice them. To accomplish this, we recommend the use of clay to make the solids and dental floss for cutting. If you have an art department that has wire cutters for clay, those are even better. You may also try plastic knives, if you have those available. We like clay because it is stiff, and if you use a thin cutting tool, clay will maintain the shape of the slice better than play dough or other softer modeling substances. We recommend that you try working with the clay yourself to see what strategies work best with your specific clay for forming crisp solids and slicing them so that the solid does not warp. This will make it easier for you to model your cutting technique for the class during the launch. You may want to provide tips to the class for forming solids and cutting smoothly based on your experience. One additional option instead of clay is kinetic sand. It is much more expensive, which is why we have not focused on it here, but it molds beautifully, with crisp edges, and cuts well.

Students may notice that slicing a prism makes two different kinds of shapes: two new three-dimensional solids and a new two-dimensional face. In this activity, we're going to focus on the two-dimensional face made by slicing. However, you'll want to be precise with your language to avoid confusion. For instance, if you simply asked, What shape is made by slicing the rectangular solid? students may try to name the new solids created, rather than the shape of the new face created.

In the closing discussion, we ask the class to consider patterns for slicing the rectangular solid to get different kinds of face shapes. An exhaustive list is not necessary, but we hope that students will discover that to create a rectangle, they can slice parallel or perpendicular to one of the rectangular faces. To make a triangular slice, you must slice through only three faces, which looks like slicing a corner off the solid. Encourage students to look for as many patterns as they can.

Activity

Launch

Launch the activity by showing students a rectangular prism made out of clay. Your rectangular prism should be large enough that students can see it well, and the faces, edges, and vertices should be as crisp as possible. Show students your cutting tool (dental floss or wire cutter), and then use it to cut the clay into two parts on an angle, so that your cut is not parallel to one face.

Without separating the two pieces, ask the class, What shape do you think the face of the slice is? Give students a chance to turn and talk to a partner about how they are visualizing the face made by slicing the solid. Invite students to come up and draw their hypotheses on the board or document camera. Ask students to explain why they believe that their shape is the one made by the slice. For each predication, ask the class whether they agree or disagree and why. The class may not come to agreement on a prediction, but do invest some time in debating predictions and how students are visualizing the slice.

Separate the two pieces of the rectangular solid to reveal the shape of the sliced face. Trace the shape of the slice onto a piece of paper on the document camera or onto the board. Discuss what you found and its relationship with students' ideas. Ask, How accurate were our predictions? Does the shape surprise you? Why or why not?

Pose the question for exploration: What different two-dimensional shapes can you make by slicing a rectangular prism?

Explore

Provide each group with clay, a cutting tool, and isometric and regular dot paper (see appendix). Ask each group to make a rectangular prism and construct a net for that prism on the regular dot paper as a record of their solid. Students explore the question, What different two-dimensional shapes can you make by slicing a rectangular...