Lesson Plan

# Volume of Cross Sections

Students will view the animated slicing and rotating of various area about the x-axis or about different lines at internet sites. Students will apply what they have learned to complete a hands-on lab activity.
Alanna G.
Classroom teacher
Gaynor McCown Expeditionary Learning School
Staten Island, United States
My Grades 9, 10, 11, 12
My Subjects Math
Objectives

Students will:

visualize the relationship between three-dimensional objects and two-dimensional representations
develop the ability to visualize movements on two- and three-dimensional objects
transition from concrete to abstract thinking
visualize the rotation of a function about the x or y-axes or about a horizontal or vertical line
take very common food items and imagine these items revolved about an axis
mentally visualize the planar region that created the three-dimensional object
explain how the two representations are connected
explain how to predict what a given cross section will look like on a graph
physically demonstrate slicing solids to create cross sections

Subjects
Math

#### 1 Hook/Attention Getter

The students should have already learned how to find area between two curves. They should also understand Riemann sums, how to integrate basic functions, and understand the fundamental theorem of calculus.

This activity allows the user to explore different cross sections of various cones, cylinders, prisms, and pyramids. Cross sections can be difficult for students to understand, given that the concept requires that students switch between three-dimensional shapes and two-dimensional representations of those shapes. While this applet was designed to help students create and visualize cross sections of different solids, some students may also benefit from hands-on activities where they can physically slice open various solids before attempting more abstract explorations such as this applet.

Student Instructions

Do Now: Use the Cross Section Flyer to answer the exploration questions below.

1. Using the Double Cone setting and the Rotate Slice slider bar, describe the
placement of the slice that results in a cross section that is a:
• Circle
• Ellipse
• Parabola
• Hyperbola

2. Using the Pyramid setting and the Lateral Faces slider bar, describe how the cross sections change as you increase the number of lateral faces.

3. Describe at least one similarity between pyramid cross sections and prism cross sections.

4. Describe at least one difference between pyramid cross sections and prism cross sections.

5. Describe at least one similarity between prism cross sections and cylinder cross sections.

6. Describe at least one difference between prism cross sections and cylinder cross sections.

#### 2 Applications of Integration

In groups of two, students will access the internet and go to the Applications of Integration site. The site will show the student step by step how to understand how the volume of a three-dimensional shape can be formed by rotating a planar region about an axis. The site goes through many examples and the students should examine each of these. The students will have clearer visual imagery of these problems than merely learning this from a text. The students should discuss the animations with each other and attempt the exercises that the site provides for them to do.

Allow students to search for other sites. Have students share sites they find that are good with each other. At the end of this 90-minute period get the whole class together to discuss what they have learned.

#### 3 Independent Practice

Activity: Creating

For the second 90-minute period have students bring in various candies. The students will complete a lab.

Student Instructions

Imagine the planar region that was revolved about either a horizontal or vertical line to create the various candies on your tray. Draw that planar region on your paper and label the drawing with the candy it represents.

Represent a typical slice on your drawing and label it with the calculus symbols as the internet site did. Hint: use ∆x and ƒ(x).

Show symbolically the volume of each of the candies by summing up all of the slices to create the volume of the candy.

If possible suggest what function could possibly be used to create each candy.

#### 4 Wrap-Up

Activity: Assessing

The goals of the first 90-minute period will be assessed both by teacher observation and also during the class discussion period each group will discuss how to find the volume of some function revolved about an axis of their choice. They must set up the problem but not actually integrate it.

The second laboratory experience for the students will be assessed through the lab report that each student will turn in at the end of the session.