# Lesson 2

Slicing Solids

## 2.1: Slice This (5 minutes)

### Warm-up

The purpose of this activity is for students to visualize what a cross section might look like and then test the prediction by observing the result of slicing through a solid. Cylindrical food items, such as cheese or carrots, are convenient examples.

### Launch

Arrange students in groups of 2. Tell students that a cross section is the intersection between a solid and a plane, or a two-dimensional figure that extends forever in all directions. Using a cylindrical food item such as cheese or carrots, or another cylindrical object, demonstrate that slicing a cylinder parallel to its base produces a circular cross section.

Then, give students quiet work time and then time to share their work with a partner.

### Student Facing

Imagine slicing a cylinder with a straight cut. The flat surface you sliced along is called a cross section. Try to sketch all the possible kinds of cross sections of a cylinder.

### Anticipated Misconceptions

Students may not consider non-horizontal or non-vertical cross sections at first. Remind them that a cross section is the intersection of any plane with a solid—the plane doesn't have to be vertical or horizontal.

### Activity Synthesis

Ask students to share their predictions of what the cross sections will look like. Demonstrate slicing each cylindrical food item according to student instructions to see several examples.

## 2.2: Slice That (20 minutes)

### Activity

In this activity, students continue to develop familiarity with three-dimensional solids and their cross sections. Students use spatial visualization to predict what cross sections might look like and then test their predictions.

This activity works best when each student has access to devices that can run the applet because students will benefit from seeing the relationship in a dynamic way. If students don’t have individual access, projecting the applet would be helpful during the synthesis.

### Launch

Arrange students in groups of 3–4. Ask students to think about definitions of some geometric solids: spheres, prisms, pyramids, cones, and cylinders. Give students some quiet work time and then time to share their work with a partner. Follow with a whole-class discussion.

A sphere is the set of points in three-dimensional space the same distance from some center. A prism has two congruent faces (or sides) that are called bases. The bases are connected by quadrilaterals. A cylinder is like a prism except the bases are circles. A pyramid has one base. The remaining faces are triangles that all meet at a single vertex. A cone is like a pyramid except the base is a circle.

Representing, Conversing: MLR7 Compare and Connect. Use this routine to help students develop the mathematical language of cross sections of geometric solids. After students explore the cross sections of their solid, invite them to create a visual display of the cross sections they found. Then ask students to quietly circulate and observe at least two other visual displays in the room. Give students quiet think time to consider what is the same and what is different about their cross sections. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify the language students use to compare and contrast various cross sections of a solid.
Design Principle(s): Cultivate conversation
Engagement: Develop Effort and Persistence. Connect a new concept to one with which students have experienced success. For example, remind students about the cross sections of a cylinder from the previous activity. Ask students how they created various cross sections of a cylinder such as a circle, rectangle, and an ellipse. Then ask students how they can apply this method to create various cross sections of their solid.
Supports accessibility for: Social-emotional skills; Conceptual processing

### Student Facing

The triangle is a cross section formed when the plane slices through the cube.

1. Sketch predictions of all the kinds of cross sections that could be created as the plane moves through the cube.
2. The 3 red points control the movement of the plane. Click on them to move them up and down or side to side. You will see one of these movement arrows appear. Sketch any new cross sections you find after slicing.

### Student Facing

#### Are you ready for more?

Delete the cube and build another solid by following the directions in its Tooltip. Make predictions about the the kinds of cross sections that could be created if the plane moves through the solid. Move your plane to confirm.

### Launch

Arrange students in groups of 3–4. Ask students to think about definitions of some geometric solids: spheres, prisms, pyramids, cones, and cylinders. Give students some quiet work time and then time to share their work with a partner. Follow with a whole-class discussion.

A sphere is the set of points in three-dimensional space the same distance from some center. A prism has two congruent faces (or sides) that are called bases. The bases are connected by quadrilaterals. A cylinder is like a prism except the bases are circles. A pyramid has one base. The remaining faces are triangles that all meet at a single vertex.  A cone is like a pyramid except the base is a circle.

Give each group clay or playdough formed into the shape of a three-dimensional solid (cube, sphere, cylinder, cone, or other solids), and dental floss to slice the clay. Tell students that to view multiple cross sections, they will slice the shape, then re-form the shape and slice again.

An alternative is to find food items with interesting cross sections or three-dimensional foam solids from a craft store and providing plastic knives to slice the solids. In this case, provide each group with several of the same solid so they can experiment with multiple slices.

Try to include a sphere, a cube, and a cone in the collection of solids.

Representing, Conversing: MLR7 Compare and Connect. Use this routine to help students develop the mathematical language of cross sections of geometric solids. After students explore the cross sections of their solid, invite them to create a visual display of the cross sections they found. Then ask students to quietly circulate and observe at least two other visual displays in the room. Give students quiet think time to consider what is the same and what is different about their cross sections. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify the language students use to compare and contrast various cross sections of a solid.
Design Principle(s): Cultivate conversation
Engagement: Develop Effort and Persistence. Connect a new concept to one with which students have experienced success. For example, remind students about the cross sections of a cylinder from the previous activity. Ask students how they created various cross sections of a cylinder such as a circle, rectangle, and an ellipse. Then ask students how they can apply this method to create various cross sections of their solid.
Supports accessibility for: Social-emotional skills; Conceptual processing

### Student Facing

1. Sketch predictions of all the kinds of cross sections that could be created from your solid.
2. Slice your solid to confirm your predictions. Sketch any new cross sections you find after slicing.

### Anticipated Misconceptions

If using the paper and pencil version of this activity and students are stuck, suggest they slice their solids at different angles and locations to see if different cross sections are generated.

### Activity Synthesis

Invite groups of students with different solids to share their list of cross sections with the class. Ask students:

• “Were there any cross sections that caught you by surprise?” (It was surprising that a cube can have cross sections that are triangles, quadrilaterals, pentagons, and hexagons.)
• “Compare and contrast the different cross sections of a sphere.” (All the cross sections were circles, but they were different sizes.)
• “How are a cube’s cross sections different from a sphere’s?” (The cube has many differently-shaped cross sections, while the sphere’s cross sections are all circles.)

## 2.3: Stack ‘Em Up (10 minutes)

### Activity

In the last activity, students started with solids and identified various cross sections. In this activity, students view three-dimensional slabs of a solid between parallel cross sections and try to determine what the original solid was. Being able to visualize the relationship between a solid and its cross sections is important to later work on Cavalieri’s Principle.

### Launch

Ask students, “What solid would a stack of all the same coins create?” Display a stack of quarters and note that it creates the shape of a cylinder. Then display, in order, a quarter, a nickel, a penny, and a dime. Ask, “What solid would a stack of coins decreasing in size create?” Make a stack with a few of each type of coin to make a solid that resembles a cone.

### Student Facing

Each question shows several parallel cross-sectional slabs of the same three-dimensional solid. Name each solid.

### Student Facing

#### Are you ready for more?

3D-printers stack layers of material to make a three-dimensional shape. Computer software slices a digital model of an object into layers, and the printer stacks those layers one on top of another to replicate the digital model in the real world.

1. Draw 3 different horizontal cross sections from the object in the image.
2. The layers can be printed in different thicknesses. How would the thickness of the layers affect the final appearance of the object?
3. Suppose we printed a rectangular prism. How would the thickness of the layers affect the final appearance of the prism?

### Activity Synthesis

Ask students to share their predictions for what solids are formed. Then display these images for all to see.

Now focus students’ attention on cross sections that are taken parallel to a solid’s base (for those solids that have bases). Ask students how cross sections can be used to differentiate between prisms and pyramids. (The cross sections of prisms taken parallel to the base are congruent to each other. The cross sections of pyramids taken parallel to the base are similar to each other.)

Speaking: MLR8 Discussion Supports. To help students respond to the questions for discussion, provide sentence frames such as: “The cross sections of _____ taken parallel to the base are _____ because.…” As students share their responses, press for details by asking students how they know that the cross sections of prisms taken parallel to the base are congruent, while the cross sections of pyramids taken parallel to the base are similar. Ask students to distinguish the meanings of geometric congruence and similarity.
Design Principle(s): Support sense-making; Optimize output (for justification)
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as: “I noticed _____ so I….”, “How do you know…?”, “That could/couldn’t be true because….”, and “I agree/disagree because….”
Supports accessibility for: Language; Social-emotional skills

## Lesson Synthesis

### Lesson Synthesis

In this lesson, students worked with three-dimensional solids and their cross sections. Here are questions for discussion:

• “How are the cross sections in this lesson different from the two-dimensional figures we looked at in the last lesson?” (In the last lesson, we rotated the two-dimensional figures to trace out a solid. The two-dimensional figures were usually an outline of half of the figure, and they had to have a relationship to the axis of rotation of the solid. Here, our cross sections cut through the entire solid, and they can come from anywhere in the solid.)
• “What kinds of applications of cross sections might we see in real life?” (There is a field of medicine called tomography that is about finding ways to get images of cross sections of people. Technologies like the CAT scan, the MRI, and the PET scan allow doctors to examine cross sections of a brain, a lung, or an injury and visualize what the three-dimensional body part looks like.)

## Student Lesson Summary

### Student Facing

In earlier grades, you learned some vocabulary terms about solid geometry: A sphere is the set of points in three-dimensional space the same distance from some center. A prism has two congruent faces (or sides) that are called bases. The bases are connected by parallelograms. A cylinder is like a prism except the bases are circles. A pyramid has one base. The remaining faces are triangles that all meet at a single vertex. A cone is like a pyramid except the base is a circle.

We often analyze cross sections of solids. A cross section is the intersection of a solid with a plane, or a two-dimensional figure that extends forever in all directions. For example, some cheese is sold in cylindrical blocks. If you stand the cheese on end and slice vertically, you will get a rectangle, as shown. This rectangle is a cross section of the cylinder.

Here are 3 more examples of cross sections created by intersecting a plane and a cylinder.

If you wanted to serve your cylindrical cheese at a party, you might cut it into several pieces, like this. The pieces are thin cylinders. They are like cross sections, but they are three-dimensional. All the cuts were made parallel to one another. By looking at the slices, or by stacking them up, you could figure out that the original shape of the cheese was a cylinder.

What if another cheese plate contained slices whose radii got bigger to a maximum size and then got smaller again? The cheese was probably in the shape of a sphere. A sphere has circular cross sections. The size of the circular cross sections increases as you get closer to the center of the sphere, then decreases past the center.