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Unit 1, Lesson 3

Slicing Solids

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Integrated Math 3

3.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

To Gather

Cylindrical food items

Activity Narrative

The purpose of this activity is for students to visualize what a cross-section might look like and then to 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, which is a two-dimensional figure that extends forever in all directions. Use a cylindrical food item, such as a cheese stick or carrot, or another cylindrical object, to demonstrate that slicing a cylinder parallel to its base produces a circular cross-section.

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

Activity

None

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.

Student Response

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Building on Student Thinking

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.

3.2

Activity

Instructional Routines

None

Materials

To Gather

Clay Dental floss

Activity Narrative

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 (MP1).

In the digital version of the activity, students use an applet to check their predictions for the cross-sections of solids. The applet allows students to move a plane through each solid to view the cross-sections. This activity works best when each student has individual 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 Activity Synthesis.

3.3

Activity

Instructional Routines

MLR8: Discussion Supports

Materials

To Gather

Coins

Activity Narrative

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.

Activity

None

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.)

to access Cool-down.

Student Lesson Summary

Here are some three-dimensional solids that may be familiar:

A sphere is the set of points, in three-dimensional space, all of which are 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.

A cross-section is the intersection of a solid with a plane, which is a two-dimensional figure that extends forever in all directions. For example, some cheese is sold in cylindrical blocks. If the cheese is placed on one end and sliced vertically, the slice will reveal a rectangle, as shown. This rectangle is a cross-section of the cylinder.

<p>A cylinder sliced down the center vertically by a rectangle.</p>

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

These pieces of cheese 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, we can figure out that the original shape of the cheese was a cylinder.

<p>Image of slices of cylindrical cheese spread out on a wooden cutting board.</p>

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 they get closer to the center of the sphere, and then decreases past the center.

<p>A sphere sliced horizontally into 7 parts. Colors from top to bottom, yellow, green, light blue, dark blue, light blue, green and yellow.</p>

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, that are all 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 solid), 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.

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, a 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 Functioning, Conceptual Processing

Activity

None

Your teacher will give your group a three-dimensional solid to analyze.

Sketch predictions of all the kinds of cross-sections that could be created from your solid.
Slice your solid to confirm your predictions. After slicing the solid, sketch any new cross‑sections that you found.

Student Response

to access Student Response.

Building on Student Thinking

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